We give the Butcher tableau and the stability function of the A-stable implicit Runge-Kutta methods considered in the numerical experiments (see also [35, Chapter IV.5]).
The two stage Gauss method (order 4):
1 The two stage Radau 1a method (order 3):
0 14 14 The two stage Lobatto 3c method (order 2):
0 12 12
Chapter 5
Conclusion and outlook : The Strang splitting for the heat equation with
absorbing boundary conditions
We observed that the standard Strang splitting method, which is formally of order of convergence two, su↵ers from a reduction of order when applied to semilinear parabolic problems with Dirichlet, Neumann or Robin boundary conditions. In Chapter 3, we pre-sented a modification of the Strang splitting method which removes the order reduction.
In Chapter 4, we showed that the Strang splitting remains second order convergent if the Crank-Nicolson scheme is used to solve the di↵usion flow, whereas a reduction of order occurs for the exact flow or other Runge-Kutta methods. We present here some ongoing work on the Strang splitting method applied to solve the heat equation with absorbing boundary conditions. In analogy to Chapter 4, we observe numerically that if the di↵usion flow is integrated with the Crank-Nicolson scheme, then the Strang splitting method is of order of accuracy two for problems with absorbing boundary conditions. Additionally, the Crank-Nicolson scheme allows us to discretize the one half fractional time derivative which appears naturally in the boundary conditions. The results which follow are a work in progress in collaboration with Christophe Besse and Gilles Vilmart [10].
5.1 The Strang splitting method for the semilinear heat equation with absorbing boundary
conditions
We consider the following initial value problem, fort2[0, T], on a bounded domain⌦2Rd with smooth boundary @⌦,
@tu(x, t) = u(x, t) +f(x, u(x, t)) in⌦⇥(0, T], u(x,0) =u0(x), (5.1)
@nu(x, t) +@t12u(x, t) = 0 on @⌦⇥(0, T], (5.2) 75
76 CHAPTER 5. CONCLUSION AND OUTLOOK where the operator is the Laplace operator in Rd and where @t12 denotes the Riemann-Liouville one half fractional derivative (see [55, Chapter 2.3]) defined as
@t12u= 1 p⇡@t
Z t 0
pu(s) t sds.
The boundary conditions @nu(x, t) +@t12u(x, t) = 0 are simple absorbing boundary condi-tions chosen to facilitate the analysis. Note that there exist several nonequivalent defini-tions of the fractional operator @t12 (see [55, Chapter 2])). The fractional operator @t12 is a linear operator defined as an extension of the usual power of derivatives @tn to fractional powers of derivatives@t↵,↵ 2R. For example, the formula for the derivative of monomials t , for all real parameter 2R\{ 1, 2, . . .}which are not a negative integer, is given by
@t12t =C t 12, withC 1
2 = 0 andC = ( +( +1)1
2). We then recover that@t12@t12 =@tand@t12@t121 = C0@t12p1t = 0.
The boundary conditions (5.2) allow us to approximate on⌦the solution of the following problem
@tu(x, t) =ˆ u(x, t) + ˆˆ f(x,u(x, t)) inˆ Rd⇥(0, T], u(x,ˆ 0) = ˆu0(x) (5.3)
kxlimk!1u(x, t) = 0 forˆ t 2(0, T], (5.4) where ˆu0 is assumed compactly supported in ⌦ and where u0, f are the restrictions of ˆ
u0,fˆon ⌦. In particular, when f = 0, the boundary conditions (5.2) are transparent in dimensiond= 1, which means, by definition, that the solutionuof the problem (5.1)-(5.2) is equal to the solution ˆu of the problem (5.3)-(5.4) restricted to ⌦. There exist more sophisticated boundary conditions that give better approximations of (5.3)-(5.4) which are similar to (5.2) (see [36, 56, 57]). We only consider (5.2) for simplicity of the analysis but we hope that the analysis can be extended to include more technical boundary conditions.
Note however that those more sophisticated boundary conditions are in general nonlinear which makes the analysis more complex.
We make the assumption that theL2(⌦) norm of the solution of the source term equa-tion,
@tu(x, t) = f(x, u(x, t)) in ⌦⇥(0, T], (5.5) decreases over time. Denoting ft the flow of the source term equation (5.5), we hence assume that for allv 2L2(⌦), the following inequality holds for all t 0,
k ft(v)kL2(⌦) kvkL2(⌦). (5.6) We shall prove that this is equivalent to the following condition (see Lemma 5.2.3): for all v 2L2(⌦), we have
Z
⌦
f(x, v(x))v(x)dx0. (5.7)
5.1. STRANG SPLITTING AND ABSORBING BOUNDARY CONDITIONS 77 Note that if we assume thatk ftˆ(v)kL2(Rd) kvkL2(Rd)for allv 2L2(Rd) and allt 0, then the solution of the problem (5.3)-(5.4) restricted to ⌦ satisfies ku(ˆ ·, t)kL2(⌦) kuˆ0kL2(⌦). We assume additionally that f : L2(⌦) ! L2(⌦) is globally Lipschitz continuous. This last condition could be relaxed but is important to require that the problem (5.1) as well as the numerical methods which follow are well posed for ⌧ small enough. We prove that the boundary conditions (5.2) are absorbing under the condition (5.6), which means, by definition, that the problem (5.1)-(5.2) satisfies ku(·, t)kL2(⌦) ku0kL2(⌦) for all t 0 (see Proposition 5.2.4).
In [6, 7, 8], in the context of the Schr¨odinger equation, or in [60], the authors suggest the following time semi-discretization of the continuous problem (5.1)-(5.2),
2vn(x) un 1(x)
1+z 1 (see [6, Section 3.1]). The positive number ⌧ > 0 is the time step and un is an approximation of u(tn) where tn = n⌧. We show, in Theorem 5.2.5, the stability of the midpoint scheme (5.8). The midpoint rule (5.8) is also appropriate to implement more technical nonlinear absorbing boundary conditions (see [42]). Note that in a practical implementation, it is advantageous to first compute vn in (5.8) and then deduce the value of un = 2vn un 1. This allows to remove one evaluation of un 1.
Another possibility is to discretize the fractional operator @
1
t2 with Pad´e approxima-tions. This is computationally advantageous since the discretized absorbing boundary conditions become local in time (see [6, 36]). However there exists no analysis analogous to Lemma 5.2.2 in the literature for Pad´e approximations on which the proofs of The-orem 5.2.5 and TheThe-orem 5.2.6 rely. We therefore choose to avoid Pad´e approximations in the present work. Note that the main source of difficulties that we encounter to prove convergence results is not the choice of the discretization which appear in (5.8). Difficulties already appear in the analysis for the continuous problem (5.1)-(5.2) (see Section 5.3).
The solution of the problem (5.8) is computationally costly since the problem is non-linear. In particular, it would be highly efficient to use a splitting method in order to solve separately the di↵usion equation and the source term equation. A question which arises is how to discretize the boundary conditions (5.2) in the splitting since they depend on the value of the solution at previous times. In this paper, we study the following Strang splitting method, where vn is the solution of
2vn(x) f⌧
78 CHAPTER 5. CONCLUSION AND OUTLOOK Hence, one step of the splitting (5.9) can be written as follows,
f ⌧
which is only linearly implicit. Indeed, denoting wn := f ⌧
2(un), we see that the sys-tem (5.11), with unknown wn, can be solved without an iterative Newton-like method, in contrast to the equation (5.8). We choose to use the Crank-Nicolson scheme (5.10), or equivalently the midpoint rule in this linear setting, over other Runge-Kutta methods, because as seen in Chapter 4, it allows to remove the order reduction of the Strang split-ting for Robin, Dirichlet or Neumann boundary conditions. We therefore hope that this property also holds for the absorbing boundary conditions (5.2). Furthermore, the dis-cretization of the one half derivative is known for multisteps methods (see [47, 48, 49]), for example the Crank-Nicolson method. We avoid multisteps methods with more than one step since they are not straightforward to be used inside a splitting method.
Under the assumption (5.6), we prove the stability of the Strang splitting (5.9) (see Theorem 5.2.6). Numerical experiments show that the splitting (5.9) gives results similar to the midpoint rule (5.8) and is second order convergent outside a neighbourhood oft= 0.
In the following results, we insist on the time semi-descretization of the problem (5.1)-(5.2). A finite element space discretization, with a mesh size h, can be included with a standard triangular inequality argument, kuh(tn)k kuh(tn) u(tn)k+ku(tn)k and kuhnk kuhn unk+kunk, where uhn is the solution of the fully discretized scheme and uh(tn) is the space semi-discretization at time tn.
Remark 5.1.1. As an extension of Remark 4.4.2 in Chapter 4, we observe that, in the specific case where f =f(x) is independent of the solution u, the Strang splitting (5.9) is equivalent to the midpoint rule (5.8). Indeed, for f = f(x), we have ft(u) =u+tf, and hence from (5.11) we obtain
un(x) un 1(x)
which is the Crank-Nicolson scheme or equivalently the midpoint rule applied to the equa-tion (5.1)-(5.2) when f =f(x).