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Invariantization of the Crank-Nicolson method for Burgers’ equation. (English) Zbl 1134.65055

Summary: A geometric technique to construct numerical schemes for partial differential equations (PDEs) that inherit Lie symmetries is proposed. The moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them. To illustrate the method, we study invariantization of the Crank-Nicolson scheme for Burgers equation. With careful choice of normalization equations, the invariantized schemes are shown to surpass the standard scheme, successfully removing numerical oscillation around sharp transition layers.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35K05 Heat equation
35A30 Geometric theory, characteristics, transformations in context of PDEs
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