Time stepping methods for singularly perturbed problems

The problem

The reaction–diffusion problem with Dirichlet boundary conditions

(ε)

with the operator ε = εΔ - c(x), c ≥ ϱ > 0, is considered. If ε ≪ 1, then there appear boundary layers that make the use of apapted meshes necessary with discretization Lε of ε.

When the Crank–Nicolson method is applied to the discretized problem, there exist maximum norm estimations of the form

(η-)

(for a Shishkin mesh on Ω = (0,1)). However, numerical computation suggests

(η+)

without restriction on the step size. The ultimate goal is to confirm this mathematically.

Procedure

A technique of [LM07] according to Kopteva delivers for the implicit Euler method an estimation of the kind (η⁺). Bu application onto the Crank–Nicolson method, one achieves (assuming sufficient smoothness)

with R(z) = (1 + z∕2)∕(1 - z∕2). Similar expressions are yielded for A stable Runge–Kutta methods. For the estimation of the term ∞, results of the semigroup theory come into play.

Theorem 1 (Brenner, Thomée, 1979) Let R be an A-acceptable rational approximation of a strongle continuous semigroup e^tA over the Banach space X. Then it is (for bounded T)

(η~)

Theorem 2 (Crouzeix, Thomée, 2001) Let the operator A generate an analytic semigroup over the Banach space X, such that especially

(R)

with θ = π∕2 - δ, δ (0,π∕2) (see figure). Then it is

Hence, the resolvent estimation (R) for Lε must be considered.

Results

For homogeneous boundary conditions, we have

• The differential operator ε fulfils (R), where M depends solely on δ and the space dimension. The proof is based upon the representation

  (Laplace transformation). Hence, the estimation of ∞ is crucial. Considering the kernel of the problem (_ε), this can be achieved using a result by Coulhon and Sikora.

  Attention! ε does not generate an analytic semigroup over L^∞(Ω).

• For the one-dimensional central difference approximation, we have

  where κ is the minimal averaged step size in space direction. In the case of boundary adapted meshes, this depends on ε.

  This yields an estimation of the form (η⁺) where C depends on ε.

Figure 1: Left hand side: The set ℂ\Σθ (shaded). Center and right hand side: Area of validity of (R) for Lε (with M = 3, c ≡ 1, ε = 10-3, N = 100) with central differences and equidistant and Shishkin mesh, respectively (numerically).

Numerical experiments

• For a one-dimensional, smooth problem with homogeneous Dirichlet boundary conditions, we observe as expected (τ² + N^-2 ln²N). Moreover, for all of the tested A-stable methods of higher order p, the maximum norm error is bounded by (τ^p + N^-2 ln²N).
• With a test problem, the exact solution of which is not continuously differentiable at x = 0 and t = 0, merely (τ¹+N^-2 ln²N) for all methods

Implementation When applying A-stable Runge–Kutta methods, in each time step a block equation system of dimension sN needs to be solved. We could show numericall that when solving it with GMRES, preconditioning with block Jacobi and block Gauss–Seidel strategies works well.

References