SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Predictor corrector algorithm with the discrete variational derivative method

D. Furihata

The discrete variational derivative method is a structure-preserving numerical method for partial differential equations. The obtained schemes mimic some variational structures of the original equations, and they inherit the conservation property or dissipation one. Those are nonlinear when the original equations are nonlinear, and this nonlinearity may be a computational difficulty. We have developed the linearization technique to overcome this difficulty, but the obtained linear schemes tend to be unstable. Furthermore, the technique is applicable to only lower order polynomial equations. In such a situation, we extended the technique to be applicable to nonpolynomial problems. This means that we are able to obtain fast schemes for every equation, but we do not improve the unstable tendency of the obtained schemes. We, therefore, attempt a breakthrough based on the classical predictor-corrector iteration method to avoid this tendency.

[1] D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-preserving Numerical Method for Partial Differential Equations, CRC Press, Florida, 2010.
[2] D. Furihata and T. Matsuo, A Stable, Convergent, Conservative and Linear Finite Difference Scheme for the Cahn-Hilliard Equation, Japan J. Indust. Appl. Math., 20 (2003), pp. 65-85.

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