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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.

65M06Finite difference methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
35K05Heat equation
35A30Geometric theory for PDE, characteristics, transformations
Full Text: DOI
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