On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms. (English) Zbl 1243.65108
Summary: The stability of nonlinear explicit difference schemes with not, in general, open domains of the scheme operators are studied. For the case of path-connected, bounded, and Lipschitz domains, we establish the notion that a multi-level nonlinear explicit scheme is stable iff the corresponding scheme in variations is stable. A new modification of the central Lax-Friedrichs scheme is developed to be of the second-order accuracy. The modified scheme is based on nonstaggered grids. A monotone piecewise cubic interpolation is used in the central scheme to give an accurate approximation for the model in question. The stability of the modified scheme is investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second-order nonstaggered central scheme based on operator-splitting techniques.
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
Keywords:
hyperbolic equations; stability; scheme in variation; numerical examples; nonlinear explicit difference schemes; central Lax-Friedrichs scheme; conservation laws; operator-splitting techniquesReferences:
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