Increasing asymptotic stability of Crank-Nicolson method Alexei A. Medovikov Vyacheslav I. Lebedev.

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Presentation transcript:

Increasing asymptotic stability of Crank-Nicolson method Alexei A. Medovikov Vyacheslav I. Lebedev

Summary The Crank-Nicolson method has second order accuracy, but for stiff ODEs, the numerical solution has unexpected oscillatory behavior, which can be explained in term of its stability function Variable time steps by Crank-Nicolson method allow us to formulate optimization problem for roots of the stability function The solution of this problem is the rational Zolotarev function We present robust algorithm of step-size selection and numerical results of the optimization procedure

The exact solution of the heat equation can be found by the method of separation of variables We expect to have similar properties from the numerical solution:

Method of lines: Lebedev, V. I. The equations and convergence of a differential-difference method (the method of lines). (Russian) Vestnik Moskov. Univ. 10 (1955), no. 10, To solve the ODE we use midpoint rule or trapezoidal rule:

Stability function of the classical Crank-Nicolson method (a)

Initial value of the heat equation (a). Exact solution of the heat equation (b). The solution of the heat equation by Crank-Nicolson method with 3 constant steps (c), and solution by the optimal method with the same sum of steps (d):

Fourier coefficients

Composition Methods ODEs generate a map: Runge-Kutta method generates a map: Composition of maps generated by RK method is composition method: Properties of RK map depend on division

Stability function Stability function of composition methods: Applying RK method to simple test problem lead to function-multiplier, which is responsible for stability of the method

How to optimize m and in order to have Composition of mid-point rules define new method and appropriate choice of steps allows us to improve properties of the stability function maximum average time-step:

Zolotarev rational function

Theorem (Medovikov): Sum of steps of Zolotarev function for the interval equals

The Algorithm Wachspress E.L. Extended application of alternating direction implicit model problem theory. SIAM J. Appl. Math. 11 (1963)

Initial value of the heat equation (a). Exact solution of the heat equation (b). The solution of the heat equation by Crank-Nicolson method with 3 constant steps (c), and solution by the optimal method with the same sum of steps (d):

Embedded methods