Thesis
Reference
The Strang Splitting Method for Semilinear Parabolic Problems with Noncanonical Boundary Conditions
BERTOLI, Guillaume Balthazar
Abstract
Splitting methods are powerful numerical schemes which allow us to divide an evolution problem into easier subproblems. For reaction-diffusion equations, the Strang splitting method, which is formally second order accurate, allows us to solve separately the diffusive process and the reaction process. In this context however, the Strang splitting suffers in general from a reduction of order which can drastically reduce its efficiency. This reduction of accuracy occurs because the reaction flow does not preserve the domain boundary conditions in general. In this thesis, we develop and analyze new remedies to avoid the reduction of order of the Strang splitting for semilinear parabolic problems with Dirichlet, Neumann, Robin or absorbing boundary conditions. Numerical experiments illustrate the behavior of these new approaches.
BERTOLI, Guillaume Balthazar. The Strang Splitting Method for Semilinear Parabolic Problems with Noncanonical Boundary Conditions. Thèse de doctorat : Univ. Genève, 2021, no. Sc. 5572
DOI : 10.13097/archive-ouverte/unige:153813 URN : urn:nbn:ch:unige-1538137
Available at:
http://archive-ouverte.unige.ch/unige:153813
Disclaimer: layout of this document may differ from the published version.
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UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES
Section de math´ematiques Dr. Gilles Vilmart
The Strang Splitting Method for Semilinear Parabolic Problems with Noncanonical Boundary Conditions
TH` ESE
Pr´esent´ee `a la Facult´e des Sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es Sciences, mention Math´ematiques
par
Guillaume Balthazar BERTOLI
de
Dardagny (GE)
Th`ese N 5572
GEN` EVE
Atelier d’impression ReproMail
2021
Remerciements
J’ai eu la chance de passer mes quatre ann´ees de th`ese dans les meilleures conditions possibles. Je le dois `a toutes les personnes qui ont ´et´e `a mes cˆot´es pendant ce parcours.
Je remercie tout d’abord Gilles Vilmart, qui s’est battu il y a quatre ans pour que je puisse commencer ce travail avec lui. Merci d’avoir ´et´e si pr´esent tout au long de ces quatre ann´ees de th`ese et de m’avoir fait d´ecouvrir la recherche dans de si bonnes conditions. Je te remercie pour toute ta g´en´erosit´e, ta patience et pour m’avoir permis de m’´epanouir dans mon travail. Je souhaite ensuite remercier tout particuli`erement Christophe Besse, avec qui j’ai eu un immense plaisir `a discuter les vendredi apr`es-midi en compagnie de Gilles. J’ai pass´e une superbe semaine `a Toulouse en ta compagnie. Merci de m’avoir si bien accueilli. Je te remercie pour tout ton support et ton ´ecoute. Un grand merci `a Laurence Halpern, Marcus Grote et Martin Gander pour avoir accept´e de faire partie de mon jury de th`ese. Je les remercie pour leur lecture attentive de ma th`ese et leurs retours.
Un grand merci `a Martin Gander pour tous ses conseils et son soutien. Un grand merci
`a Joselle Besson pour m’avoir aid´e tout au long de ces quatre ann´ees concernant tous les probl`emes administratifs. Merci `a Peter Pasche pour le maintien des machines linux.
Je remercie Adrien et Fathi pour la pi`ece de th´eˆatre sur la th´eorie des nœuds que l’on a pr´epar´ee et jou´ee ensemble. Ce fut une superbe exp´erience qui m’a beaucoup enrichie.
Je souhaite remercier toute l’´equipe du Mathscope pour tout ce que vous m’avez appris.
Un grand merci `a Tommaso, Giancarlo et Ding qui ont fait de notre ”full rank office” un lieu de travail dans lequel j’ai eu grand plaisir `a travailler. Je souhaite remercier Marco, Ibrahim, Jhih-Huang, Thibaut, Conor, Michal, Nicolas, Ausra, Renaud, Eugene, Bastien, Pablo, Christine, Aitor, Eiichi, Elise, Sandie, Bart pour toutes ces discussions au Z-bar, ces caf´es que l’on a partag´es ensemble et pour avoir rendu ces ann´ees de travail pleines de bons moments.
Un immense merci `a Ir`ene pour ton ´energie in´epuisable que tu me transmets chaque jour. Je remercie mes parents pour tout leur support et pour m’avoir permis d’arriver jusqu’ici. Un grand merci `a mes fr`eres pour tout leur soutien. A vos cˆot´es, je ne saurais oublier que tout est possible, tout est permis.
iii
Abstract
Splitting methods are powerful numerical schemes which allow us to divide an evolu- tion problem into easier subproblems. For reaction-di↵usion equations, the Strang splitting method, which is formally second order accurate, allows us to solve separately the di↵usive process and the reaction process. In this context however, the Strang splitting su↵ers in general from a reduction of order which can drastically reduce its efficiency. This reduc- tion of accuracy occurs because the reaction flow does not preserve the domain boundary conditions in general.
In Chapter 1, we provide a concise introduction to splitting methods applied to semi- linear parabolic problems. We explain the di↵erent possible choices to define the Strang splitting method. We then introduce absorbing boundary conditions for the heat equation, which are an important motivation of our work. We explain the role of the boundary conditions in the reduction of order which occurs for the Strang splitting method and we present an existing modification technique to avoid this reduction of order. We then briefly explain the main results and the main ideas developed in this thesis with more details in the subsequent chapters.
In Chapter 2, we explain, with a simple one dimensional example, how one can estimate the global error of the Strang splitting with the help of semigroup theory. A key ingredient is to introduce in the analysis quadrature formulas which are not necessarily involved in the scheme themselves. We show that classical error estimates based on Taylor expansion cannot be used since the quadrature error is a function of an unbounded operator. Instead, we use the parabolic smoothing property of the analytic semigroups to find an appropriate estimate. We therefore show that a second order estimate of the Strang splitting is easily obtained formally, whereas a rigorous analysis only allows to show a fractional order of con- vergence between one and two. In the second part of this chapter, we present Runge-Kutta methods and we explain the link between the rational stability function of A-stable and L-stable Runge-Kutta methods and the rational approximations of analytic semigroups.
In Chapter 3, we present a new modification of the Strang splitting method which allows us to recover the order two of accuracy. This new modification, compared to the existing modification presented in Chapter 1, allows us to factorize the splitting similarly to the classical Strang splitting using the semigroup property of the exact flows. Furthermore, no additional evaluation of the source term or di↵usion term is required, which reduces the computational cost of the method. We prove that the new modified splitting is second order accurate and thus has no order reduction and we perform numerical experiments to compare the new modification to the existing modification presented in Chapter 1.
Numerical experiments suggest that the new modified splitting method remains accurate even if the source term is sti↵.
In Chapter 4, we discuss an unexpected interplay between the Strang splitting method and the Crank-Nicolson scheme. We show that if the di↵usion flow is approached with the Crank-Nicolson scheme, then the Strang splitting is second order convergent outside a neighbourhood of the origin t = 0, when in comparison reduction of order occurs for the exact flow or other classical Runge-Kutta methods. We prove this property in the specific case where the source term f = f(x) is independent of the solution, in which case the Strang splitting with Crank-Nicolson is equivalent to the Crank-Nicolson scheme applied
iv
to the whole problem. Numerical experiments suggest that this convergence property also holds if the source termf =f(x, u) is nonlinear.
In Chapter 5, we present an ongoing work on the Strang splitting method applied to the heat equation with absorbing boundary conditions. We prove the stability of the Strang splitting method, where the di↵usion is solved with the Crank-Nicolson scheme, under the condition that the L2(⌦) norm of the solution of the source term equation decreases over time. We conjecture that the Strang splitting is second order accurate outside a neigh- bourhood of the origint = 0 when applied to the heat equation with absorbing boundary conditions. Numerical experiments suggest that the convergence results for the Strang splitting method with the Crank-Nicolson scheme, developed in Chapter 4 in the con- text of Dirichlet, Neumann or Robin boundary conditions, persist for absorbing boundary conditions.
v
R´ esum´ e
Les m´ethodes de splitting sont des sch´emas num´eriques qui permettent de diviser un probl`eme d’´evolution en plusieurs probl`emes auxiliaires plus simples `a int´egrer. Pour les
´equations de type r´eaction-di↵usion, le splitting de Strang, qui est formellement d’ordre de convergence deux, permet de r´esoudre s´epar´ement les processus de di↵usion et de r´eaction.
Dans ce contexte, le splitting de Strang sou↵re en g´en´eral d’une r´eduction d’ordre qui peut drastiquement r´eduire son efficacit´e. Cette perte de pr´ecision apparaˆıt car le probl`eme auxiliaire de r´eaction ne pr´eserve en g´en´eral pas les conditions au bord du domaine.
Dans le chapitre 1, nous introduisons de mani`ere concise les m´ethodes de splitting appliqu´ees aux probl`emes semilin´eaires paraboliques. Nous expliquons les di↵´erents choix possibles pour d´efinir le splitting de Strang. Nous introduisons ensuite les conditions au bord absorbantes pour l’´equation de la chaleur qui sont une motivation importante de notre travail. Nous expliquons le rˆole des conditions au bord dans la r´eduction d’ordre du splitting de Strang et nous pr´esentons une technique de modification existante pour
´eviter cette r´eduction d’ordre. Nous expliquons ensuite bri`evement les r´esultats et les id´ees principales d´evelopp´ees plus en d´etail dans les chapitres suivants.
Dans le chapitre 2, nous expliquons, `a l’aide d’un exemple en une dimension, com- ment estimer l’erreur globale du splitting de Strang `a l’aide de la th´eorie des semigroupes.
Un ingr´edient clef consiste `a introduire dans l’analyse des formules de quadrature qui n’apparaissent pas n´ecessairement dans le sch´ema num´erique. Nous montrons que les esti- mations classiques de l’erreur bas´ees sur les s´eries de Taylor ne peuvent pas ˆetre utilis´ees car l’erreur de quadrature est une fonction d’un op´erateur non born´e. Nous utilisons `a la place la propri´et´e de r´egularisation des semigroupes analytiques pour trouver une esti- mation appropri´ee. Nous d´emontrons ainsi qu’une estimation d’ordre deux est facilement obtenue formellement alors qu’une analyse rigoureuse de l’erreur ne permet que de mon- trer un ordre fractionnaire de convergence entre un et deux. Dans la seconde partie de ce chapitre, nous pr´esentons les m´ethodes de Runge-Kutta et nous expliquons le lien entre la fonction de stabilit´e rationnelle des m´ethodes de Runge-Kutta A-stables et L-stables et les approximations rationnelles des semigroupes analytiques.
Dans le chapitre 3, nous pr´esentons une nouvelle modification du splitting de Strang qui permet de retrouver l’ordre de convergence deux. Cette nouvelle modification, compar´ee `a la modification existante pr´esent´ee au chapitre 1, permet de factoriser le splitting modifi´e similairement au splitting classique en utilisant la propri´et´e de semigroupe des flots exacts.
De plus, aucune ´evaluation du terme source ou du terme de di↵usion n’est n´ecessaire, ce qui r´eduit le coˆut de la m´ethode. Nous d´emontrons que le nouveau splitting modifi´e est d’ordre de convergence deux et donc ne sou↵re pas de r´eduction d’ordre. Nous pr´esentons des exp´eriences num´eriques qui comparent la nouvelle modification `a la modification existante pr´esent´ee au chapitre 1. Les exp´eriences num´eriques sugg`erent que le nouveau splitting modifi´e reste pr´ecis mˆeme si l’´equation du terme source est raide.
Dans le chapitre 4, nous discutons une interaction inattendue entre le splitting de Strang et le sch´ema de Crank-Nicolson. Nous montrons que si le flot de la di↵usion est approch´e avec le sch´ema de Crank-Nicolson, alors le splitting de Strang est d’ordre de convergence deux en dehors d’un voisinage de l’origine t = 0. En comparaison, une r´eduction d’ordre du splitting apparaˆıt si le flot de la di↵usion est approch´e par une autre m´ethode de Runge-
vi
Kutta ou si le flot exact de la di↵usion est utilis´e. Nous d´emontrons cette propri´et´e dans le cas particulier o`u le terme source f =f(x) est ind´ependant de la solution. Dans ce cas particulier, le splitting de Strang pour lequel la di↵usion est approch´ee avec le sch´ema de Crank-Nicolson est ´equivalent au sch´ema de Crank-Nicolson appliqu´e au probl`eme complet.
Les exp´eriences num´eriques sugg`erent que cette propri´et´e de convergence persiste si f d´epend de la solution.
Dans le chapitre 5, nous pr´esentons notre travail en cours sur le splitting de Strang ap- pliqu´e `a l’´equation de la chaleur avec des conditions au bord absorbantes. Nous d´emontrons la stabilit´e du splitting de Strang sous l’hypoth`ese que la norm L2(⌦) de la solution de l’´equation du terme source d´ecroˆıt au cours du temps. Nous conjecturons que le splitting de Strang est d’ordre de convergence deux en dehors d’un voisinage de l’origine t = 0 lorsque il est appliqu´e `a l’´equation de la chaleur avec des conditions au bord absorbantes.
Les exp´eriences num´eriques sugg`erent en e↵et que, pour le probl`eme avec des conditions au bord absorbantes, le splitting de Strang se comporte ´egalement sans r´eduction d’ordre,
`a l’instar du cas ´etudi´e au chapitre 4 o`u des conditions au bord de type Dirichlet, Neuman ou Robin sont prescrites.
Contents
1 Introduction and main results 1
1.1 Splitting methods for semilinear parabolic problems . . . 1
1.1.1 Choice of the splitting method . . . 2
1.1.2 Absorbing boundary conditions . . . 3
1.1.3 Reduction of order of the Strang splitting . . . 7
1.1.4 Existing modification to avoid the order reduction . . . 8
1.2 New remedies to avoid order reduction phenomena . . . 9
1.2.1 A new modification of the Strang splitting . . . 9
1.2.2 The Strang splitting with Crank-Nicolson . . . 11
2 Preliminaries 17 2.1 Discussion on our approach . . . 17
2.1.1 Exact solution and numerical solution . . . 17
2.1.2 Error estimate . . . 19
2.2 Stability of Runge-Kutta methods . . . 20
3 Five parts modified Strang splitting 25 3.1 Introduction . . . 25
3.2 Analytical framework . . . 27
3.3 Description of the method . . . 28
3.4 Convergence analysis . . . 29
3.4.1 Quadrature error analysis . . . 30
3.4.2 Order one error estimate for the five parts Strang splitting . . . 35
3.4.3 Analysis of the corrector function . . . 39
3.4.4 Order two error estimate for the five parts modified Strang splitting 40 3.5 Numerical experiments . . . 43
4 Strang splitting with Crank-Nicolson 49 4.1 Introduction . . . 49
4.2 Analytical framework . . . 51
4.3 The splitting with Crank-Nicolson . . . 53
4.4 Convergence analysis forf =f(x) . . . 55
4.4.1 Main results . . . 56
4.4.2 Preliminaries . . . 57
4.4.3 Local error . . . 59
4.4.4 Global error . . . 60 vii
viii CONTENTS
4.5 Numerical experiments . . . 61
4.5.1 Solution independent source term . . . 62
4.5.2 Solution dependent source term . . . 65
4.5.3 Stationary problems . . . 65
4.6 Crank-Nicolson approximation of a semigroup . . . 68
4.7 Runge Kutta method Butcher tableaus . . . 74
5 Conclusion and outlook 75 5.1 Strang splitting and absorbing boundary conditions . . . 75
5.2 Stability results . . . 78
5.2.1 Auxiliary lemma . . . 78
5.2.2 Stability of the continuous problem . . . 79
5.2.3 Stability of the implicit mid-point method . . . 80
5.2.4 Stability of the Strang splitting method . . . 81
5.3 Numerical experiments . . . 83
5.4 Conclusion . . . 85
Bibliography 87
List of Figures 92
Chapter 1
Introduction and main results
In this chapter, we introduce splitting methods in the context of semilinear parabolic prob- lems. We recall why a standard implementation of the Strang splitting method, formally of order of accuracy two, su↵ers in general from a reduction of order due to the presence of boundary conditions. We present an existing modification to avoid the reduction of order.
We then briefly explain the main new results and the main ideas of this thesis presented in Chapter 3 and Chapter 4.
1.1 Splitting methods for semilinear parabolic problems
We are interested in the numerical solution of di↵usion-reaction equations using splitting methods. Let ⌦ be a bounded open subset of Rd, d 1, with smooth boundary @⌦ and let the time t belong to the interval [0, T]. We consider the semilinear parabolic problem
@tu(x, t) = Du(x, t) +f(x, u(x, t)) in⌦⇥(0, T], (1.1) together with boundary conditions and an initial condition u0(x). The di↵usive operator Dis a linear operator and the source term f is a smooth function, possibly nonlinear. The goal of a splitting method is to divide the main problem (1.1) into auxiliary subproblems which are easier to solve than the main problem (1.1). There exists a variety of ways to split the problem (1.1). We consider here the source term problem, defined as
@tu(x, t) =f(x, u(x, t)) in ⌦⇥(0, T], (1.2) and the di↵usion problem,
@tu(x, t) = Du(x, t) in ⌦⇥(0, T]. (1.3) The boundary conditions are only prescribed for the di↵usion problem (1.3) since the source term equation (1.2) is an ordinary di↵erential equation which, with an initial condition, has a unique solution. The exact flows of the equations (1.2) and (1.3) for a timet are denoted by ft and Dt . A splitting method consists in solving successively the subproblems (1.3) and (1.2) in order to find an approximation of the main problem (1.1). The simplest
1
2 CHAPTER 1. INTRODUCTION AND MAIN RESULTS splitting method is called the Lie splitting. For a time step ⌧ > 0, one step of the Lie splitting is defined as follows,
un+1 = D⌧ f⌧(un),
where un denotes the approximation of u(tn) at time tn = n⌧. Alternatively, one could invert the flows f⌧ and D⌧ and obtain un+1 = f⌧ D⌧ (un). The Lie splitting is formally a method of order of accuracy one. By formally, we mean that if, instead of (1.1), we consider the ordinary di↵erential equation @tu(t) = Du(t) + f(u(t)) on Rd with some initial condition u0 2 Rd, where D is a matrix and f a smooth function, then we have k( D⌧ f⌧)n(u0) u(tn)k C⌧ with C independent of ⌧, n and tn = n⌧ T. The Lie splitting is rarely used in practice since there exists another splitting called the Strang splitting which is more accurate in general. One step of the Strang splitting method with a time step⌧ is defined as
un+1 = f⌧ 2
D
⌧ f
⌧
2(un). (1.4)
Alternatively, the flows f⌧ and D⌧ can be inverted, un+1 = D⌧
2
f⌧ D
⌧
2(un). (1.5)
1.1.1 Choice of the splitting method
It is not always clear in the literature which of the splitting methods (1.5) and (1.4) should be used to approximate problems of the form (1.1). We shall focus our research on the splitting (1.4). Indeed, the flows which appear in the Strang splitting (1.4) are rarely solved exactly in practice and we observe in Chapter 4 that the splitting (1.4) has remarkable properties when the solution of the di↵usion problem (1.3) is approximated with the Crank- Nicolson scheme. Additionally, the fact that the flow f⌧
2 is computed before the di↵usion flow D⌧ is essential in the modification technique we present in Chapter 3.
The Strang splitting is formally a method of order of accuracy two, which means, in the context of ordinary di↵erential equations on Rd, k( f⌧
2
D⌧ f⌧
2)n(u0) u(tn)k C⌧2 with C independent of ⌧, n and tn T. In the context of PDEs, this second order error estimate holds also in some specific cases in the L2(⌦) norm. In general, however, this estimate holds only for ⌧1+↵ instead of ⌧2 with 0 ↵ 1. This is a central question of this thesis explained with more details in Section 1.1.3.
Due to the semigroup property of the exact flows, f⌧ 2
f
⌧
2 = f⌧, we observe that the Strang splitting requires in practice only two evaluations of flows at each step of the algorithm,
un= f⌧
2
D
⌧ ( f⌧ D⌧ )n 1 f⌧
2(u0). (1.6)
This makes the Strang spitting as cheap as the Lie splitting but with greater accuracy.
There exist splitting methods of arbitrary high order defined as
un+1 = D↵1⌧ f1⌧ . . . D↵m⌧ fm⌧(un), (1.7)
1.1. SPLITTING METHODS FOR SEMILINEAR PARABOLIC PROBLEMS 3 with i 2Rand↵i 2Rfor alli= 1, . . . , m. However, it was proved in [30] that any splitting method of the form (1.7) with an order of convergence strictly higher than two satisfies mini=1,...,m↵i <0 and mini=1,...,m i <0 (see also [13]). When the solution of a semilinear parabolic equation of the form (1.1) is approximated with a splitting method, one needs to solve successively the problems of (1.3) and (1.2). Since the di↵usion equation is unstable in the negative time direction, it makes splittings of the form (1.7), with order higher than two, unusable for such equations. To build higher order splitting methods to solve di↵usion-reaction equations of the form (1.1), a possibility is to use complex coefficients
i 2C and ↵i 2C for all i= 1, . . . , mwith positive real part (see [15, 19, 38]) or to use a linear combination of second order splitting methods of the form (1.7) (see [21]). We shall not consider the above techniques to build higher order splitting methods and we focus our research on the Strang splitting method which is, for the above reason, the most widely used splitting method in the context of parabolic semilinear equations. For more details on splitting methods, see [33, 39, 52].
1.1.2 Absorbing boundary conditions
Splitting methods for semilinear parabolic problems of the form (1.1) are often studied with simple boundary conditions, for example periodic or homogeneous Dirichlet boundary conditions. In practice, it is often impossible to avoid more technical boundary conditions.
For example, let us consider the problem
@tu(x, t) =ˆ u(x, t) + ˆˆ f(x,u(x, t)) inˆ Rd⇥(0, T], u(x,ˆ 0) = ˆu0(x)
kxk!1lim u(x, t) = 0 forˆ t2(0, T], (1.8)
where is the Laplace operator on Rd. Let us assume that the initial condition ˆu0 is compactly supported in a bounded domain ⌦⇢Rd. Assume that we are interested in the restriction of the solution of (1.8) to ⌦. We consider the problem
@tu(x, t) = u(x, t) +f(x, u(x, t)) in⌦⇥(0, T], u(x,0) =u0(x), (1.9) were u0 and f are the restriction of ˆu0 and ˆf to ⌦. We make the assumption that the L2(⌦) norm of the solution of the source term equation (1.2) decreases over time: for all t 0 and for all v 2L2(⌦), we have
k ft(v)kL2(⌦) kvkL2(⌦). (1.10) We shall prove that this is equivalent to the following assumption onf: for allv 2L2(⌦), the inequalityR
⌦f(x, v(x))v(x)0 holds. Note that under the assumption that the source term ˆf satisfies k ftˆ(v)kL2(Rd) kvkL2(Rd) for allv 2L2(Rd) and t 0, the exact solution ˆ
u(t, x) restricted to⌦satisfiesku(ˆ ·, t)kL2(⌦) kuˆ0kL2(⌦). We aim at constructing boundary conditions on @⌦which make the solution uof (1.9) equal to the solution ˆu of (1.8) on⌦.
Such conditions are called transparent boundary conditions and are very difficult to find in general, particularly for nonlinear source terms. Instead, we use absorbing boundary conditions, which make the solutionuof (1.9) an approximation of ˆuon⌦. More precisely, boundary conditions for the problem (1.9) are called absorbing if, for those boundary conditions, the solution u(x, t) of (1.9) satisfies ku(·, t)kL2(⌦) ku0kL2(⌦) for all t 0,
4 CHAPTER 1. INTRODUCTION AND MAIN RESULTS similarly to the solution of the problem (1.8) restricted to⌦. One of the simplest choices is to prescribe
@nu(x, t) +@t12u(x, t) = 0 on @⌦⇥(0, T], (1.11) where @t12 denotes the Riemann-Liouville one half fractional derivative (see [55, Chapter 2.3]) defined as
@t12u(x, t) = 1 p⇡@t
Z t 0
u(x, s) pt sds.
As an example, we consider the following one dimensional problem on R,
@tu(x, t) =ˆ @xxu(x, t)ˆ u(x, t)ˆ 3 in R⇥(0,2], u(x,ˆ 0) = e x2,
kxlimk!1u(x, t) = 0 forˆ t 2(0,2], (1.12) whose initial conditions ˆu(x,0) is numerically compactly supported in the bounded domain
⌦ = ( L, L) with L = 6. The solution of (1.12) for t > 0 has R itself as support and it is difficult in general to discretize the whole domain R. As a remedy, we consider the following problem on the bounded domain ⌦= ( L, L),
@tu(x, t) = @xxu(x, t) u(x, t)3 in ⌦⇥(0,2], u(x,0) = e x2. (1.13) We would like to find boundary conditions which make the solution of (1.13) a good approximation of the exact solution of (1.12) restricted to ⌦. As a first approximation, we choose to prescribe homogeneous boundary conditions, u( L, t) = u(L, t) = 0, to the problem (1.13). As seen in Figure 1.1a, the solution of (1.13) with the boundary conditions u( L, t) = u(L, t) = 0, has an error close to 10 2 on the boundary of the domain ⌦ and a minimal error close to 10 7 inside of ⌦. We then prescribe the absorbing boundary conditions (1.11). We observe that the solution is much more accurate for the absorbing boundary conditions (1.11) and the error varies between 10 7 and 10 12.
In the specific case where f = 0, the boundary conditions (1.11) are transparent in dimension d= 1, which means, by definition, that the solution u of (1.9), with boundary conditions (1.11), is exactly equal to the restriction of the solution ˜uof (1.8) on ⌦. When f is not the zero function, there exist more sophisticated absorbing boundary conditions which give better results than (1.11) that we do not consider here (see [36, 56, 57]). In Chapter 5, we prove that the boundary conditions (1.11) are absorbing under the assump- tion (1.10).
To compute an approximation of (1.9) with absorbing boundary conditions (1.11), it is suggested in [60] or in [6, 7, 8], in the context of the Schr¨odinger equation, to use the implicit midpoint method (which is equivalent to the Crank-Nicolson scheme if f is an affine function of u),
2vn+1(x) un(x)
⌧ = vn+1(x) +f(x, vn+1(x)) in ⌦, (1.14)
1.1. SPLITTING METHODS FOR SEMILINEAR PARABOLIC PROBLEMS 5
-8 -6 -4 -2 0 2 4 6 8
x 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y
t=0 t=1 t=2
(a)Graph of the solution
-6 -4 -2 0 2 4 6
x 10-10
10-8 10-6 10-4 10-2
Error
(b)Absorbing and Dirichlet conditions Figure 1.1: On the left picture, we draw the solution of the unbounded problem(1.12)at time t= 0, t= 1 and t= 2. On the right picture, we search boundary conditions on @⌦, which make the solution of the problem (1.9) an approximation of the solution of the unbounded problem (1.8) restricted to ⌦. We observe that the absorbing boundary conditions (1.11) give a better approximation than homogeneous Dirichlet boundary conditions.
wherevn+1 = un+12+un andv0 =u0. For this time semi-discretization, the one half derivative
@t12 can be discretized and we obtain
@nvn+1(x) + r2
⌧ Xn+1
k=0
n+1 kvk(x) = 0 on@⌦, (1.15)
where the series ( k)k 0 is the inverse Z-transform of q+
1 z 1
1+z 1 (see [6, Section 3.1]). For more details on the discretization of@t12, see [47, 48, 49]. In Chapter 5, we prove the stability of the midpoint method (1.14) with boundary conditions (1.15) under the hypothesis (1.10).
There exist few studies which try to apply splitting methods to solve problems of the form (1.9) with absorbing boundary conditions. In [9], the Strang splitting applied to the Schr¨odinger equation with absorbing boundary conditions is studied for the first time.
More recently, in [27], the authors study a splitting method applied to a linear dispersive equation with transparent boundary conditions. Although we do not consider absorbing boundary conditions in Chapter 3 and 4, it was an important motivation of our research to develop a second order Strang splitting method to solve the problem (1.9) with boundary conditions similar to (1.11). The complexity of the nonlocal boundary conditions (1.11) and the high computational cost required to handle those conditions make the splitting method (1.5) particularly interesting. The Strang splitting method removes indeed the necessity to use a Newton like iterative algorithm to solve the implicit equation resulting from the discretization (1.14). In Chapter 5, we present some ongoing work on this topic,
6 CHAPTER 1. INTRODUCTION AND MAIN RESULTS
10-3 Time step 10-2 10-8
10-6 10-4 10-2
Error
Midpoint StrangCN
(a)Absorbing and Dirichlet conditions
10-3 Time step 10-2 10-8
10-6 10-4 10-2
Error
Midpoint StrangCN
(b)Midpoint rule and Strang splitting
Figure 1.2: (Chapter 5, Figure 5.2) We observe that the Strang splitting (1.16), denoted StrangCN, is second order convergent when applied to the one dimensional nonlinear prob- lem (1.13)with absorbing boundary conditions (1.11)or when applied to the problem (1.18) and gives results similar to the midpoint rule approximation (1.14). Reference slopes of or- der one and two are given in dashed lines.
where we study the Strang splitting method given by un+1 = f⌧
2
Dn,CN
⌧
f⌧
2(un). (1.16)
The numerical flow Dt n,CN( f⌧
2(un)) is defined as the Crank-Nicolson approximation, or equivalently the midpoint approximation for this linear setting, of the di↵usion prob- lem (1.3) with absorbing boundary conditions (1.11). More precisely, we have Dtn,CN( f⌧
2(un)) = 2vn+1 f⌧
2(un), where vn+1 is the solution of 2vn+1(x) f⌧
2(un(x))
⌧ = vn+1(x) in⌦
@nvn+1(x) + r2
⌧ Xn+1
k=0
n+1 kvk(x) = 0 on @⌦. (1.17) Note that thevkwhich appear in (1.17) are hence given byvk= 12( f⌧
2(uk 1) + f ⌧
2(uk)). In Chapter 5, we prove the stability of the Strang splitting (1.16) under the assumption (1.10).
Numerical experiments show that the splitting method (1.16), when it is applied to the problem (1.9) with absorbing boundary conditions (1.11), is second order convergent out- side a neighbourhood oft >0, similarly to the results we present in Chapter 4 for Dirichlet, Neumann and Robin boundary conditions. For example, as seen in Figure 1.2, we observe that the Strang splitting (1.16), denoted StrangCN, applied to the problem (1.13) with absorbing boundary conditions (1.11) or applied to the problem
@tu(x, t) = @xxu(x, t) + sin(u(x, t)) in ⌦⇥(0, T], u(x,0) = e x2,
@nu(x, t) +@t12u(x, t) = 0 on @⌦⇥(0, T], (1.18) is second order convergent and gives results similar to the midpoint approximation (1.14).
1.1. SPLITTING METHODS FOR SEMILINEAR PARABOLIC PROBLEMS 7
10-3 Time step 10-2 10-10
10-8 10-6 10-4 10-2
Error
(a) Di↵usion solved exactly
10-3 Time step 10-2 10-10
10-8 10-6 10-4 10-2
Error
(b) Di↵usion solved with Crank-Nicolson Figure 1.3: The Strang splitting (1.4) is of order of convergence two when applied to the problem (1.20) but a reduction of order occurs for the equation (1.19). When the Crank- Nicolson method is used to approximate the solution of the di↵usion problem (1.3), then the reduction of order is avoided. Reference slopes of order one and two are given in dashed lines.
1.1.3 Reduction of order of the Strang splitting
In contrast to the non sti↵ ODE case, when applied to PDEs, we observe that the Strang splitting is in general not second order convergent when it is used to approximate the solution of semilinear parabolic problems of the form (1.1) (see [39]). Often, it has only a reduced fractional order of convergence between one and two (see [25, Section 4.3]). For example, let us consider, fort2[0,0.1], the two following parabolic problems on⌦= (0,1), with Dirichlet and Robin boundary conditions, respectively,
@tu(x, t) = @xxu(x, t) + 1 in (0,1)⇥(0, T], u(0, t) =u(1, t) = 1, u(x,0) = 1 (1.19) and
@tu(x, t) = @xxu(x, t) + e x in (0,1)⇥(0, T], @xu(0, t) +u(0, t) = @xu(1, t) +u(1, t) = 1,
u(x,0) = 1. (1.20)
As seen in Figure 1.3a, the Strang splitting (with exact flows) is second order convergent when applied to the equation (1.20) but it has only an order of convergence close to one when applied to the equation (1.19). This reduction of order is the result of an undesired interaction between the reaction flow f⌧
2 and the boundary conditions that are prescribed in (1.1) and in (1.3).
The boundary conditions are omitted in the formal computation of the second order convergence of the Strang splitting but they play a fundamental role in the analysis. The boundary conditions are defined as
Bu(x, t) =b(x) on @⌦⇥(0, T], (1.21) where the linear operatorB represents Dirichlet, Neumann or Robin boundary conditions, which respectively correspond to Bu=u,Bu =@nu and Bu=↵u+ @nu, with↵, 6= 0.
8 CHAPTER 1. INTRODUCTION AND MAIN RESULTS The function b(x) is a smooth function on @⌦. When dividing the main problem (1.1) into the source term problem (1.2) and the di↵usion problem (1.3), the boundary condi- tions (1.21) are only prescribed for the di↵usion problem (1.3). In particular the boundary conditions are in general not preserved by the flow f⌧
2. As a consequence, the flow Dt is no longer di↵erentiable for t = 0. This causes the reduction of order of the Strang splitting method. For example, in (1.20), the flow of f(x) = e x preserves the Robin boundary conditions, that is, ifun satisfies (1.21), then f⌧
2(un) preserves (1.21). The flow of f(x) = 1 however does not preserve Dirichlet boundary conditions. This explains why the Strang splitting is of order of accuracy two when applied to the problem (1.20) but has only order close to one when applied to the problem (1.19). In general, if f satisfies Bf(x, u(x, t)) = 0 on @⌦, then the boundary conditions are preserved. In the literature, reduction of order is often associated to inhomogeneous boundary conditions. This is be- cause for chemical reactions, it is natural to havef(x,0) = 0, that is, no reactant yields no reaction, whose flow therefore preserves homogeneous Dirichlet boundary conditions. In theory, it is however possible to have a reduction of order of the Strang splitting method even if homogeneous boundary conditions are prescribed. For instance, the problem (1.19) with u(0, t) = u(1, t) = 0 and u(0, x) = sin(⇡x) yields similar results to Figure 1.3a. Our goal is to develop new tools to avoid the reduction of order we observe in Figure 1.3a. A new modification technique is developed in Chapter 3.
In Figure 1.3b, we approximate the di↵usion flow D⌧ which appears in the splitting (1.4) with the Crank-Nicolson scheme. In the literature, splitting methods are often studied with exact flows only (see [12, 24, 25]). In practice however, the problems (1.2) and (1.3) are rarely solved exactly. They are instead approximated with numerical methods. Standard numerical methods for the problems (1.2) and (1.3) are Runge-Kutta methods (see [34, Chapter 2]). We observe in Figure 1.3b that the reduction of order is completely avoided if the Crank-Nicolson scheme is used to approximate the di↵usion flow D⌧ which appears in the Strang splitting (1.4). This unexpected interaction between the Crank-Nicolson scheme and the Strang splitting (1.4) is the subject of Chapter 4.
1.1.4 Existing modification to avoid the order reduction
In [24] and [25], a modification of the Strang splitting (1.5) is given which permits to avoid the reduction of order. The idea is to split the main equation (1.1) between a modified source term problem
@tu(x, t) = f(x, u(x, t)) qn(x) in ⌦⇥(0, T], and a modified di↵usion problem
@tu(x, t) = Du(x, t) +qn(x) in ⌦⇥(0, T], Bu(x, t) =b(x) on@⌦⇥(0, T].
The functionqn is a smooth function constructed at each step of the splitting such that f⌧ almost preserves the boundary conditions. More precisely, qn is constructed to satisfy the following boundary conditions,
Bqn=Bf(un) on@⌦. (1.22)
1.2. NEW REMEDIES TO AVOID ORDER REDUCTION PHENOMENA 9 In the interior of the domain, the functionqncan be chosen freely as long as it is sufficiently smooth. The modified splitting presented in [25] and [24] is then given by
un+1 = D+q⌧ n
2
f qn
⌧
D+qn
⌧
2 (un). (1.23)
It is shown in [24, 25] that the modified Strang splitting (1.23) is second order convergent for semilinear parabolic problems of the form (1.1) with boundary conditions (1.21). In [23], several options to construct qn are given. A challenge is to construct a function qn which minimizes the error constant for a low computational cost.
A disadvantage of the modified splitting (1.23) is that in general it cannot be factorized as in (1.6). The modified splitting (1.23) therefore requires in practice two evaluations of the di↵usion flow D+q⌧ k
2 and one evaluation of the reaction flow f q⌧ k at each step of the algorithm,
un=
nY1 k=0
( D+q⌧ k 2
f qk
⌧
D+qk
⌧
2 )(u0).
In addition, (1.22) may require the computation of the derivative f0(un) in the case of Robin or Neumann boundary conditions. In the specific case whereqn =q is independent ofn, which is the case iff =f(x) does not depend onu, we can use the semigroup property of D+q⌧
2 and obtain
un = D+q⌧
2
f q
⌧ ( D+q⌧ f q⌧ )n 1 D+q⌧
2 (u0).
We refer to [3], where another technique is presented to avoid the order reduction, which consists in introducing intermediate boundary conditions in the di↵usion equation (1.3) which appear in the Strang splitting (1.4). We refer to [4, 26] for a comparison between the modification presented in [24, 25] and the technique explained in [3].
1.2 New remedies to avoid order reduction phenomena
1.2.1 A new modification of the Strang splitting
In Chapter 3, we present a new modification technique which allow to remove the order reduction of the splitting (1.4) (see [12]). The advantage of this new modified splitting is that it can always be factorized similarly to (1.6). Furthermore, numerical experiments show that this new modified splitting is more stable than the splitting (1.23) if the source term equation is sti↵. This new modification technique is inspired by symmetric projection methods on manifolds (see [33, Chapter V.4.1]). The idea is to apply a projection ⌧qn
2 after the source term flow f⌧
2 such that ⌧qn
2
f⌧
2 satisfies almost the boundary conditions (1.21).
The new modified splitting is given by un+1 = f⌧
2
qn
⌧ 2
D+qn
⌧
qn
⌧ 2
f
⌧
2(un). (1.24)
10 CHAPTER 1. INTRODUCTION AND MAIN RESULTS The projection ⌧qn
2 is defined as the exact flow of the equation
@tu(x, t) = qn(x) in ⌦⇥(0, T], whose solution is given by ⌧qn
2 ( f⌧
2(un)) = f⌧
2(un) ⌧2qn. We prove the following theorem, valid for a class of nonlinear source terms f(u), which states that the modified Strang splitting (1.24) is second order convergent.
Theorem 1.2.1. (Chapter 3, Theorem 3.4.1) The splitting (1.24) satisfies the bound kun u(tn)kL2(⌦) C⌧2 0n⌧ T,
for all ⌧ small enough, and where the constant C is independent of⌧ and n and tn. The function qn is constructed such that
Bqn = 2
⌧(B f⌧
2(un) b) on @⌦. (1.25)
Note that the right term in (1.25) is a finite di↵erence approximation of Bf(un), 2
⌧(B f⌧
2(un) b)⇡ 2
⌧(B f⌧
2(un) Bun)⇡Bf(un), since f⌧
2(un) = un+ ⌧2f(un) +O(⌧2). Hence the condition (1.25) is close to the condi- tion (1.22) for smoothf. Similarly to the modification in the splitting (1.23), the conditions that the functionqnneeds to satisfy are only prescribed on the boundary. In the interior of the domain ⌦, the function qn is constructed to be smooth. Through the splitting (1.24), the function qn is constructed directly after the computation of f⌧
2(un). No additional evaluation off is therefore required. This is an advantage iff is computationally costly or iff is large. Indeed the flow f⌧
2 is usually smoother than the source term f and numerical experiments show that the modified splitting (1.24) remains accurate when the source term equation (1.2) is sti↵, in contrast to the splitting (1.23). For example, let us consider the following two dimensional problem on⌦= [0,1]2 with t2[0,0.1],
@tu(x, y, t) =@xxu(x, y, t) +@yyu(x, y, t) + (1 Msin(⇡x) sin(⇡y))u(x, y, t)2, u(0, y, t) = 1 + ey
2 , @nu(1, y, t) = e
2, @nu(x,0, t) = 1
2, @nu(x,1, t) = e 2, u(x, y,0) = ex+ ey
2 . (1.26)
As seen in Figure 1.4, whenM = 1, the modified splitting methods (1.23) and (1.24) give results that are quite similar. When M = 100 however, the splitting (1.24) is a lot more accurate than the splitting (1.23), which even diverges for ⌧ >10 2.
Another advantage of the splitting (1.24) is that, in practice, it requires only one evaluation of the di↵usion flow D⌧ and one evaluation of the source term flow f⌧ at each step, similarly to the classical splitting (1.4). Indeed, we have
un = f⌧ 2
qn 1
⌧ 2
D+qn 1
⌧
qn 1
⌧ 2
nY2 k=0
( f⌧ ⌧qk 2
D+qk
⌧
qk
⌧
2 ) f⌧
2(u0).
Since the functionsqkare cheap to construct and since the flows ⌧qk are cheap to compute, this makes the splitting (1.24) almost as inexpensive as the classical splitting (1.4) but with no order reduction.
1.2. NEW REMEDIES TO AVOID ORDER REDUCTION PHENOMENA 11
10-3 10-2 10-1
Time step 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5
Nonsti↵caseM = 1.
10-3 10-2 10-1
Time step 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5
Sti↵case M = 100.
Figure 1.4: (Chapter 3, Figure 3.3) When applied to the problem (1.26)with M = 1, both modified splitting methods (1.23), denoted StrangM3, and (1.24), denoted StrangM5, per- form better than the classical splitting (1.4), denoted Strang. For M = 100, that is, when the source term becomes sti↵, the splitting (1.24) is more stable than the splitting (1.23) which is less accurate than the classical splitting (1.4). Reference slopes of order one and two are given in dashed lines.
1.2.2 The Strang splitting with Crank-Nicolson
In Chapter 4, we study the classical splitting (1.4) where the exact flow D⌧ is approximated with the Crank-Nicolson scheme. As seen in Figure 1.3b, using the Crank-Nicolson scheme to approximate the solution of the di↵usion problem (1.3) allows to remove the order reduction of the classical splitting (1.4) for the one dimensional problem (1.19). Note that here, we do not use the modification techniques (1.24) or (1.23). A question which naturally arises is whether or not this property is specific to the Crank-Nicolson scheme or if it holds for other Runge-Kutta methods. Let us consider again the problem (1.19). We use the Strang splitting (1.4) on this problem and we try to approximate the di↵usion part (1.3) with several Runge-Kutta methods. We use the two stage Gauss method, the two stage Radau 1a method and the two stage Lobatto 3c method. Note that these Runge-Kutta methods have order of accuracy equal or higher than two. As seen in Figure 1.5a, only the Strang splitting (1.4) with Crank-Nicolson is of order two. The other Runge-Kutta methods give a result similar to or worse than the splitting (1.4) where the di↵usion flow is solved exactly, denotedStrangEXP. In Figure 1.5b, we observe that the Crank-Nicolson scheme does not allow to remove the order reduction of the splitting (1.5) nor does any of the considered Runge-Kutta methods.
We denote D,CN⌧ the Crank-Nicolson approximation of the di↵usion flow D⌧ . One step of the Strang splitting with the Crank-Nicolson scheme is given by
un+1 = f⌧
2
D,CN
⌧
f⌧
2(un), (1.27)
where D⌧ in (1.4) has been replaced by D,CN⌧ . We prove that, for the specific case where f =f(x) does not depend on the solutionu, the splitting (1.27) is second order convergent
12 CHAPTER 1. INTRODUCTION AND MAIN RESULTS
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
StrangEXP StrangGauss StrangRadau StrangCN StrangLobatto
(a) un+1= f⌧ 2
D⌧ f
⌧ 2(un).
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
StrangEXP2 StrangGauss2 StrangRadau2 StrangCN2 StrangLobatto2
(b)un+1 = D⌧ 2
f⌧ D
⌧ 2(un).
Figure 1.5: (Chapter 4, Figure 4.2) We solve the problem (1.19) with the Strang split- ting (1.4), where we approximate the di↵usion flow with several Runge-Kutta methods. The Crank-Nicolson scheme is the only Runge-Kutta method which removes the order reduction.
For the splitting (1.5), no Runge-Kutta method allows to remove the order reduction. Ref- erence slopes of order one and two are given in dashed lines.
outside a neighbourhood of the origin t = 0. More precisely, we prove the following theorem.
Theorem 1.2.2. (Chapter 4, Theorem 4.4.3) Let en = un u(tn) be the global error of the splitting method (1.27) applied to the parabolic problem (1.1)-(1.21) with f = f(x) independent of u. Let u0 2 H2(⌦) satisfying Bu0 = b on @⌦. Then, for all n 0, the global error en is given by
en = r(⌧A)n en⌧A A 1(Du0+f) (1.28) and it satisfies the bound
kenkL2(⌦) C⌧2 tn
, (1.29)
where C is a constant independent of ⌧, n, and tn=n⌧. The operator A is the restriction of the operator D to the domain D(A) = {u 2 H2(⌦) ; Bu = 0 on @⌦}, the set of functions with homogeneous boundary conditionsBu(x) = 0 on @⌦. The rational function r(z) = (1 +z2)/(1 z2) is the stability function of the Crank-Nicolson scheme.
Although it is assumed in Theorem 1.2.2 that f = f(x) does not depend on the so- lution u, numerical experiments suggest that the splitting with Crank-Nicolson (1.27) is also second order accurate outside a neighbourhood of t = 0 for a source term f which is solution dependent. For example, let us consider the following two dimensional problems
1.2. NEW REMEDIES TO AVOID ORDER REDUCTION PHENOMENA 13
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
StrangEXP StrangCN
(a) f(u) =u with Robin boundary conditions.
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
StrangEXP StrangCN
(b)f(u) =u2 with mixed boundary conditions.
Figure 1.6: (Chapter 4, Figure 4.3) When applied on the two dimensional problems (1.30) and (1.31), the Strang splitting with Crank-Nicolson (1.27)is second order convergent when in comparison the splitting with exact flows (1.4) su↵ers from a reduction of order.
on⌦= (0,1)2 with t2[0,0.1],
@tu(x, y, t) =@xxu(x, y, t) +@yyu(x, y, t) +u(x, y, t), u(0, y, t) +@nu(0, y, t) =y2, u(1, y, t) +@nu(1, y, t) =y2+ 2, u(x,0, t) +@nu(x,0, t) =x2, u(x,1, t) +@nu(x,1, t) =x2 + 2,
u(x, y,0) =x2+y2. (1.30)
and
@tu(x, y, t) =@xxu(x, y, t) +@yyu(x, y, t) +u2(x, y, t),
@nu(0, y, t) = 1
2, u(1, y, t) = e1+ ey 2 ,
@nu(x,0, t) = 1
2, u(x,1, t) = ex+ e1 2 , u(x, y,0) = ex+ ey
2 . (1.31)
In Figure 1.6, we observe that the splitting (1.27) with Crank-Nicolson is second order convergent for the problems (1.30) and (1.31) and thus has no order reduction. In com- parison, the splitting (1.4) with exact flows has a fractional order of convergence between one and two.
To deduce the estimate (1.29) from the formula (1.28), we use the following result from [37] (see also Section 4.6 where a self-contained proof of this result is presented),
Theorem 1.2.3. [37, Theorem 2.1] Let A and r(y) be defined as in Theorem 1.2.2. Let u2D(A). Then we have
k r(⌧A)n en⌧A ukL2(⌦) C⌧2
tn kAukL2(⌦), with C independent of u, ⌧, n, and tn=n⌧.
14 CHAPTER 1. INTRODUCTION AND MAIN RESULTS In general, when solving the equation resulting from the space discretization of a dif- fusive problem of the form (1.1), one needs to carefully choose the appropriate method.
Indeed, the ODE resulting from the space discretization is a sti↵ problem that cannot be solved efficiently with all Runge-Kutta methods. In particular, standard explicit method are in general ill suited to integrate in time such problems because they have a restricted domain of stability. To avoid instability, the time step then needs to be sufficiently small compared to the mesh size. This is the so called Courant–Friedrichs–Lewy condition. To avoid a too restrictive condition, a possibility is to use explicit stabilized methods which are designed to have a large stability domain (see [1, 2]). For linear problems however, implicit schemes are often a common choice in comparison to nonlinear problems, and A-stable implicit schemes allow to completely avoid the Courant–Friedrichs–Lewy condi- tion. Runge-Kutta methods are called A-stable if their stability domain contains allz2C with negative real part. A-stability is not always sufficient and sometimes L-stability is required. A method is called L-stable if it is A-stable and if its stability function R(z) satisfies lim|z|!1R(z) = 0 (see [35, Chapter IV.3] for more details). The Crank-Nicolson scheme is a method which is A-stable but not L-stable. Theorem 1.2.3 is an unexpected re- sult since classical second order estimates for A-stable methods requireu2D(A2) (see [44, Theorem 4.2]). Indeed L-stability is usually required for second order estimates outside a neighbourhood oft = 0 with only u2Lp(⌦) or u2D(A) (see [44, Theorem 4.4]).
The Crank-Nicolson scheme, directly applied to the main problem (1.1) with boundary conditions (1.21), is defined as
un+1 un
⌧ =Dun+1+un
2 +f(un+1) +f(un)
2 in⌦,
Bun+1+un
2 =b on @⌦. (1.32)
If f is nonlinear, this requires, at each step of the Crank-Nicolson method, the use of an iterative algorithm to find the solutionun+1. An advantage of the Strang splitting (1.27) is that it allows to apply the Crank-Nicolson scheme directly to the linear di↵usion equa- tion (1.3), which completely avoids the necessity to use an iterative algorithm as in (1.32).
To better compare the Strang splitting (1.27) with the Crank-Nicolson scheme (1.32), we observe that the splitting (1.27) can be written as
f
⌧
2(un+1) f⌧ 2(un)
⌧ =D
f
⌧
2(un+1) + f⌧ 2(un)
2 in⌦,
B
f
⌧
2(un+1) + f⌧
2(un)
2 =b on@⌦, (1.33)
where we directly see that the Strang splitting allows us to avoid the nonlinear part of the scheme (1.32).
As a direct consequence of the convergence result for the splitting (1.27), we prove the following corollary, which states that the Crank-Nicolson scheme (1.32) is second order convergent outsidet = 0 when applied to the problem (1.1) withf =f(x) andu0 2H2(⌦) satisfying Bu0 =b on@⌦.
Corollary 1.2.4. Let en = un u(tn) be the global error of the Crank-Nicolson scheme applied to the whole parabolic problem (1.1) with f = f(x) independent of u. Let u0 2
1.2. NEW REMEDIES TO AVOID ORDER REDUCTION PHENOMENA 15 H2(⌦) satisfyingBu0 =b on@⌦. Then, for alln 0, the global error en is given by (1.28) and it satisfies the bound
kenkL2(⌦) C⌧2 tn
.
Indeed, when f =f(x), the splitting (1.27) is equivalent to the Crank-Nicolson scheme applied to the whole problem (1.1). To see this, we observe that, if f is independent of u, then (1.33) is equivalent to the Crank-Nicolson scheme (1.32). Hence, as a direct consequence of Theorem 1.2.2, we deduce Corollary 1.2.4. Note that if f =f(u) depends nonlinearly on the solution u itself, the equivalence of (1.32) and (1.33) does not hold in general.
