1.1 Splitting methods for semilinear parabolic problems
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
(a)Graph of the solution
-6 -4 -2 0 2 4 6
(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) + 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
(a)Absorbing and Dirichlet conditions
10-3 Time step 10-2
(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(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)) = 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.