Abstract
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. The selection strategy consists of the following two stages. At the first stage, we employ explicit calculations for selecting the appropriate time step. At the second stage, using the implicit scheme, we calculate the solution at a new time level. This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side. The same formulas for the evaluation of the time step are obtained by using a comparison of two approximate solutions, which are obtained using the explicit scheme with the primary time step and the step that is reduced by half. Numerical results are presented for a model parabolic boundary value problem, which demonstrate the robustness of the developed algorithm for the time step selection.
Highlights
In numerically solving boundary value problems for time-dependent equations, emphasis is on discretizations in time [1,2,7]
We propose a new technology how to select the time step, which focuses on the approximate solution of boundary value problems for parabolic equations
Taking into account the stiffness of a problem, it is based on finding the approximate solution at the new time level using the unconditionally stable implicit approximations in time
Summary
In numerically solving boundary value problems for time-dependent equations, emphasis is on discretizations in time [1,2,7]. In numerically solving the Cauchy problem for systems of ordinary differential equations, there are are applied nested methods, where two approximate solutions of different orders of accuracy are compared. We propose a new technology how to select the time step, which focuses on the approximate solution of boundary value problems for parabolic equations. For such problems, we apply the implicit approximation in time of first (or rarely second) order.
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