Adaptive energy preserving methods for partial differential equations. (English) Zbl 1395.65062
Summary: A framework for constructing integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The approach can be used with both finite difference and partition of unity methods, thereby including finite element methods. The schemes are then extended to accommodate \(r\)-, \(h\)- and \(p\)-adaptivity. To illustrate the ideas, the method is applied to the Korteweg-de Vries equation and the sine-Gordon equation. Results from numerical experiments are presented.
MSC:
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
energy preserving methods; partition of unity method; finite difference method; adaptive methods; moving mesh methodReferences:
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