Improvement of Stability and Accuracy of Time-Evolution Equation by Implicit Integration
Keywords:Time-evolution equation, Implicit Method, Stability, Accuracy
The stability and accuracy of the numerical integration of the time-evolution equation obtained by discretizing an unsteady partial differential equation with respect to space variables has a crucial importance in solving the unsteady partial differebtial equation numerically.
The second and fourth Runge-Kutta mrthods are widely used in the numerical calculation. However, in some cases, the stability is not sufficient.
New implicit methods are proposed to increase stability and accuracy of the solution of the time-evolution-equation. Three new implicit methods, that is, implicit method usng linear approximation (IMP1), one using parabolic approximation (IMP2) and one using cubic approximation (IMP3) are proposed. In the case of linear problem, IMP1 is identical to the implicit method by Crank and Nicholson.
The stability of various methods including Runge-Kutta method is discussed theoretically and numerically, and the numerical examples are shown to show the effectiveness of the implicit methods. It is proposed that the most practical way to increase both the accuracy and the stability in the solution of boundary value problems may be to use IMP1 and the smaller spatial mesh size.
J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Camb. Phil. Soc., (1947) Vol.43, pp. 50-67.
H. H. Rosenbrock, Some general implicit processes for the numerical solution of differential equations, The Computer Journal (1963) 5 (4), pp. 329-330.
Burgers' equation, http://en.wikipedia.org/wiki/Burgers'_equation.
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