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A universal solver for hyperbolic equations by cubic-polynomial interpolation. I: One-dimensional solver. (English) Zbl 0991.65521
Summary: A new numerical method is proposed for general hyperbolic equations. The scheme uses a spatial profile interpolated with a cubic polynomial within a grid cell, and is described in an explicit finite-difference form by assuming that both a physical quantity and its spatial derivative obey the master equation. The method gives stable and less diffusive results even without any flux limiter. It is successfully applied to the KdV equation, a one-dimensional shock-tube problem and a cylindrically converging shock wave.
For Part II see Comput. Phys. Commun. 66, No. 2-3, 233–242 (1991; Zbl 0991.65522).

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
35L45 Initial value problems for first-order hyperbolic systems
76M25 Other numerical methods (fluid mechanics) (MSC2010)
35Q53 KdV equations (Korteweg-de Vries equations)
35L67 Shocks and singularities for hyperbolic equations
65D05 Numerical interpolation
Full Text: DOI
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