Let us consider an ordinary di↵erential equation on RN, N 2N, of the form d
dty(t) = f(t, y(t)) for t2[0, T], y(0) =y0, (2.11) wheref : [0, T]⇥RN !RN is a smooth vector field (sufficiently di↵erentiable). We assume that the problem (2.11) has a unique solution for allt 2[0, T]. Runge-Kutta methods are standard numerical schemes for the time integration of problems of the form (2.11). One step of ans stage Runge-Kutta method with a time step ⌧ >0 is defined as
ki =f tn+ci⌧, yn+⌧
2.2. STABILITY OF RUNGE-KUTTA METHODS 21 with aij, bi, ci 2 R and tn = n⌧. Here the ki also depend on n, but this is omitted for brevity of the notations. If aij = 0 for all j i, the method is called explicit. If not, the method is implicit and a possibly nonlinear system of equations has to be solved at each step n. We assume that ⌧ 0 is suffiently small, in order to make sure that there exists a unique solution for those nonlinear systems of equations which appear for implicit methods.
The existence and uniqueness are then a consequence of the fixed point theorem. We also assume the time step ⌧ is constant and does not vary with n. Runge-Kutta methods can be represented by tables called Butcher tables which contain the coefficients aij, bi, ci of the method,
c1 a11 . . . a1s
... ... ... ...
cs as1 . . . ass b1 . . . bs
The order of convergence of a Runge-Kutta method is defined as follows.
Definition 2.2.1. A Runge-Kutta method is of local order p, if for all smooth vector fields f, it satisfies
ky1 y(⌧)k C⌧p+1,
with C independent of ⌧ assumed small enough. The value y1 ⇡ y(⌧) is the numerical solution after one step of the problem (2.11).
A Runge-Kutta method of local order p is of global order p for the problem (2.11) if the vector fieldf is sufficiently smooth. By definition, a Runge-Kutta method is of global order pfor the problem (2.11) if it satisfies
kyn y(tn)k C⌧p, with C independent of⌧, n and tn =n⌧.
In the specific case where
f(t, y(t)) = y(t), 2R, (2.12)
corresponding to the Dahlquist scalar test equation (see [35, Definition IV.2.1]), a Runge-kutta method gives the following approximation after n time steps with stepsize ⌧,
yn=R( ⌧)ny0.
The function R:C!C is called the stability function of the Runge-Kutta method, from which we define the stability domain of the method (see [35, Chapter IV.3]).
Definition 2.2.2. The stability domainS of a Runge-Kutta method with stability function R is given by
S ={z 2C;|R(z)|1}.
22 CHAPTER 2. PRELIMINARIES For a Runge-Kutta method with stability functionS, if⌧ 2S, the numerical solution of the problem (2.12) stays bounded for alln 2N,|yn||R( ⌧)|n|y0||y0|. When facing a sti↵ ordinary di↵erential equation, the stability domain plays an important role for the convergence property of Runge-Kutta methods. In particular A-stable or even L-stable methods are sometimes required to have good convergence properties. For example, let us consider the parabolic problem (2.1). After a space discretization with finite elements or finite di↵erences, an explicit Runge-Kutta method is constraint by a condition of the form ⌧ < Ch2 to remain stable, where h is the mesh size. This condition is called the Courant–Friedrichs–Lewy condition. To avoid this condition, one can use an A-stable implicit Runge-Kutta method, whose stability domain contains all complex numbers z with negative real part in their stability domain.
Definition 2.2.3. A Runge-Kutta is called A-stable if its stability domain contains the half plane C = {z 2 C;<(z) 1}. If in addition limz!1R(z) = 0, then the method is called L-stable.
Hence, the numerical solution of an A-stable Runge-Kutta method, applied to the problem (2.12) with 2C , remains bounded for alln 2Nand all time steps⌧, similarly to the exact solution of the problem,y(t) = et y0, which remains bounded for all timet 0.
The rational stability functionR(z) is a rational approximation of the exponential ez. More precisely, if the Runge-Kutta method is of global order p, then R(z) is an approximation satisfying
|R(z) ez|C|z|p+1 for|z| small enough.
If A is the infinitesimal generator of a uniformly bounded analytic semigroup etA on a real (or complex) separable Banach space X with norm k·k, then the rational function R(⌧A)n is an approximation of the semigroup en⌧A. More precisely we have the following theorem.
Theorem 2.2.4. ([44, Theorem 4.2 and Theorem 4.4]) Let R be the stability function of an A-stable Runge-Kutta method of local orderp. Then foru2D(Ak)with kp, we have
k(R(⌧A)n e⌧nA)uk C⌧kkAkuk,
with C independent of n,⌧, u and tn. If the Runge-Kutta method is additionally L-stable, then we have
k(R(⌧A)n e⌧nA)uk C ⌧p
tp kkuk, (2.13)
with C independent of n,⌧, u and tn.
The Crank-Nicolson scheme is an A-stable Runge-Kutta method, of order of accuracy two, given by
0 0 0
1 1/2 1/2 1/2 1/2
yn+1 =yn+⌧f(yn) +f(yn+1)
2 .
2.2. STABILITY OF RUNGE-KUTTA METHODS 23 As proved in Section 4.6, the Crank-Nicolson scheme has the remarkable property that it satisfies the estimate (2.13) with p = 2 and k = 1 although it is not an L-stable method (see also [37, Theorem 2.1] ). This property is at the heart of the results we present in Chapter 4. For more details on Runge-Kutta methods and their stability domain, we refer to [34, 35].
Runge-Kutta methods have a long history which dates back to the mid eighteenth cen-tury with Euler works. In the late nineteenth cencen-tury, the development of a tree formalism to express elementary di↵erentials allowed us to perform algebraic analysis of numerical methods (see [17, 18, 34]). This algebraic formalism gives new insight on the behaviour of numerical methods. For example, order conditions for Runge-Kutta methods and a characterization of symplectic methods, which play an important role in the long-time in-tegration of Hamiltonian systems, are derived (see [33, Chapter VI]). The development of modern computers allows us to numerically integrate ODEs (2.11) for long periods of time.
In general, for long time integration, it is important to choose carefully approriate meth-ods in order to remain accurate. As an example, the explicit and implicit Euler methmeth-ods are both inaccurate for long time integration of Hamiltonian systems since they do not preserve the energy of the system, which is preserved by the exact flow (see [33, Chapter I.2]). Geometric numerical integration aims at building numerical methods which share geometric properties with the exact flow. The Crank-Nicolson method is an example of a symplectic method, which preserves almost the energy of Hamiltonian systems and is therefore a good candidate for the long time integration of such problems. Splitting meth-ods of the form (1.7), which can have high order of accuracy for ODEs, are other examples of symplectic methods in the context of Hamiltonian systems (see [33, Theorem VI.3.7]).
For more details on the geometric numerical integration, we refer to [14, 16, 18, 33, 45].
The splitting method described in Chapter 3 is inspired by symmetric projection meth-ods on manifolds (see [33, Chapter IV.4 and Chapter VI.4]). Let us consider the (N m)-dimensional submanifold of RN given by
M={y 2RN; g(y) = 0},
whereg :RN !Rm is smooth. We consider a problem of the form (2.11) with the property that the solution stays on the manifold Mfor all time t,
y0 2M implies y(t)2Mfor all t 0. (2.14) This property is equivalent to : g0(y)f(y) = 0 for all y 2 M. In general, the numerical solution of a Runge-Kutta method applied to a problem of the form (2.11) with the prop-erty (2.14) does not stay on the manifold M. As a remedy, we compose the Runge-Kutta method with a projection. One step of the standard projection method, from yn 2M to yn+1 2M, is given by the following algorithm.
Algorithm 2.2.5 (Standard projection method for (2.11)-(2.14), [33, Algorithm IV.4.2]).
ˆ
yn+1 = ⌧(yn) (One step of a Runge-Kutta method applied to the problem (2.11)).
Compute yn+1 defined as the projection of yˆn+1 onto the manifold M.
Algorithm 2.2.5 is simple to implement but it destroys the symmetry of symmetric Runge-Kutta methods, which are well suited for the long time integration of ⇢-reversible
24 CHAPTER 2. PRELIMINARIES problems. An alternative to Algorithm 2.2.5 is the symmetric projection method. For any symmetric Runge-Kutta method ⌧, one step of the symmetric projection method, from yn 2M toyn+1 2M, is given by the following algorithm.
Algorithm 2.2.6(Symmetric projection method for (2.11)-(2.14), [33, Algorithm V.4.1]).
ˆ
yn=yn+rg(yn)µ (perturbation step).
ˆ
yn+1 = ⌧(ˆyn) (One step of a symmetric Runge-Kutta method applied to (2.11)).
yn+1 = ˆyn+1+rg(yn+1)µ, with µ2RN such that yn+1 2M (projection step).
The structure of Algorithm 2.2.6 is similar to the modified Strang splitting method (1.24) applied to the semilinear parabolic problem (1.1) with boundary conditions (1.21). One step of the Strang splitting (1.24), fromun to un+1 is given by the following algorithm.
Algorithm 2.2.7 (Modified Strang splitting (1.24) for the PDE (1.1)-(1.21) ).
Compute wn = f⌧
2(un) (source term equation (1.2)).
Compute wˆn =wn ⌧
2qn, with qn such that Bwˆn⇡b (projection step).
Compute vn = D+q⌧ n( ˆwn) (di↵usion equation (1.3)).
Compute ˆvn =vn ⌧
2qn (perturbation step).
Compute un+1= f⌧
2(ˆvn)(source term equation (1.2)).
We observe the similarity between the step two of Algorithm 2.2.7, which is a projection onto the space of function satisfying the appropriate boundary conditions, and the step three of Algorithm 2.2.6 which is a projection onto the manifold M. The role of the step four of Algorithm 2.2.7 and of the step one of Algorithm 2.2.6 is to keep the symmetry of the respective method.
Chapter 3
Strang splitting method for semilinear parabolic problems with inhomogeneous boundary conditions: a correction based
on the flow of the nonlinearity
This chapter is identical to the paper [12] in collaboration with Gilles Vilmart.
3.1 Introduction
In this chapter, we consider a parabolic di↵erential equation of the form
@tu=Du+f(u) in⌦, Bu=b on@⌦, u(0) =u0, (3.1) where D is a linear di↵usion operator and f is a nonlinearity. A natural method for approximating (3.1) are splitting methods. The idea is to divide the main equation (3.1) into two auxiliary subproblems (3.4) and (3.5) so one can use specific numerical methods for both subproblems to enhance the global efficiency of the computation of (3.1). Let N 2N and let ⌧ = NT be the time step. Then, one step of the classical Strang splitting is either
un+1 = D⌧
2
f
⌧ D
⌧
2(un) (3.2)
or alternatively
un+1= f⌧
2
D
⌧ f⌧
2(un), (3.3)
where ft(u0) is the flow after time t of
@tu=f(u), u(0) =u0, (3.4)
and Dt (u0) is the flow after time t of
@tu=Du in ⌦, Bu=b on @⌦, u(0) =u0. (3.5) 25
26 CHAPTER 3. FIVE PARTS MODIFIED STRANG SPLITTING The Strang splitting, when applied to ODE with a sufficiently smooth solution, is a method of order two. However, when the Strang splitting is applied to solve the problem (3.1), a reduction of order can be observed in the case of non homogeneous boundary conditions as shown in [24, 25]. The reason is that Bu is not left invariant through the flow ft and therefore leaves the domain of D, which creates a discontinuity at t = 0 in the flow Dt . In this case the Strang splitting has in general a fractional order of convergence between one and two [25, section 4.3]. In [24, 25], a modification of the Strang splitting is given to recover the order two. The main idea in [25] is to find a function qn such that Bu is now left invariant by f qt n, the exact flow of
@tu=f(u) qn. (3.6)
One step of the modified splitting in [25] is then un+1 = D+q⌧ n
Numerically, one can choose any smooth function qn such that
Bqn=Bf(un) +O(⌧) on@⌦. (3.9) Several options to construct qn are presented in [23]. One challenge is then to find a correctionqn that is both cheap to compute and minimizes the constant of error.
In this chapter, we give a new modification of the classical Strang splitting that removes the order reduction and requires only one evaluation of the di↵usion flow and one evaluation of the source term flow at each step of the algorithm and allows us a cheaper construction of qn and a simpler implementation. As illustrated in the experiments, this new construction performs better for the case of a sti↵ reaction. The idea is to leaveBu unpreserved at the boundary through the flow f⌧ and then apply a correction qn afterwards that brings back the solution to the domain ofD. This new splitting is then
S⌧(un) = ( f⌧ correctionqn is now constructed from the output of the flow f⌧/2(un) and not directly from the nonlinearity f. Note that the computation of ⌧qn
2 and qn requires no evaluation of f, which is particularly useful when f is costly. More importantly, in many situations, the flow f⌧/2(un) is smoother than the nonlinearity f itself, which can avoid the possible instability due to the eventual sti↵ness of the reaction.
In section 2, we give the appropriate framework for the convergence analysis of this modified splitting. In section 3, we describe the new modification we consider in this paper. In section 4, we prove that the method is of global order two under the hypotheses made in section 2. In section 5, we present some numerical experiments to illustrate the performance of the new approach.