1.2 New remedies to avoid order reduction phenomena
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where D⌧ in (1.4) has been replaced by D,CN⌧ .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
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
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
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.
Chapter 2
Preliminaries
In this chapter, we recall from the known literature, with a simple one dimensional parabolic problem, how to estimate the error of the Strang splitting (1.4). In particular, we show that the quadrature error must be estimated carefully because it is a function of an unbounded operator.
In the second part of this chapter, we give a brief introduction to Runge-Kutta methods.
We insist on the stability of those methods which plays an important role in this thesis and we explain the link with the rational approximations of analytic semigroups.