Variable time steps optimization of $$L_{\omega}$$-stable Crank-Nicolson method.(English)Zbl 1080.65077

The heat equation in the domain $$[0,1]\times[0,\infty)$$ is approximated, using the Crank-Nicolson method, by a system of nonlinear equations. The authors propose an algorithm which uses the Crank-Nicolson method with variable time steps $$h_i$$. The special choice of the sequence of steps increases the average step size $$\tau$$ of this method while preserving the property of asymptotical stability. The choice of steps use the Zolotarev rational function which is similar of the stability function $$R(z)$$ of the Runge-Kutta method.
Numerical experiments are achieved for the heat equation with concrete initial and boundary conditions.

MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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