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