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Bound-preserving high-order schemes. (English) Zbl 1368.65149
Abgrall, Rémi (ed.) et al., Handbook on numerical methods for hyperbolic problems. Applied and modern issues. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-63910-3/hbk; 978-0-444-63911-0/ebook). Handbook of Numerical Analysis 18, 81-102 (2017).
Summary: For the initial value problem of scalar conservation laws, a bound-preserving property is desired for numerical schemes in many applications. Traditional methods to enforce a discrete maximum principle by defining the extrema as those of grid point values in finite difference schemes or cell averages in finite volume schemes usually result in an accuracy degeneracy to second-order around smooth extrema. On the other hand, successful and popular high-order accurate schemes do not satisfy a strict bound-preserving property. We review two approaches for enforcing the bound-preserving property in high-order schemes. The first one is a general framework to design a simple and efficient limiter for finite volume and discontinuous Galerkin schemes without destroying high-order accuracy. The second one is a bound-preserving flux limiter, which can be used on high-order finite difference, finite volume and discontinuous Galerkin schemes.
For the entire collection see [Zbl 1364.65001].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume 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
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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