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A flux-based modified method of characteristics. (English) Zbl 0765.76057
Summary: A flux-based modified method of characteristics (MMOC) methodology in 1D is described which has the following properties: unconditional stability (though explicit), exact answers for integer \(CFL\) (Courant) numbers, completely conservative (locally and globally) and able to utilize various flux limiters and various characteristic- (trajectory-) tracking algorithms. The use of characteristics based on cell-wise constant characteristic velocities results in considerable code simplification, and van Leer’s MUSCL [B. van Leer, J. Comput. Phys. 32, No. 1, 101- 136 (1979)] is an acurate and cost-effective flux limiter. For \(CFL>1\) the flux limiter is applied only to the non-integer part of \(CFL\), whereas the integer part is exact for constant velocities; therefore acuracy improves with larger \(CFL\). It is not a cheap algorithm, although explicit, because the operation count per time step increases with the integer part of \(CFL\), but it is much more acurate than the commonly used implicit upstream differencing. This flux-based MMOC method is well suited for groundwater flow calculations in which large local Courant numbers arise owing to grid clustering.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
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