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

Medovikov, A. A.; Lebedev, V. I.

Russ. J. Numer. Anal. Math. Model. 20, No. 3, 283-303 (2005).

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.

Reviewer: Ivan Secrieru (Chişinău)

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

Keywords:

Crank-Nicolson method; heat equation; algorithm; variable time steps; asymptotical stability; Runge-Kutta method; numerical experiments