Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

Abstract

), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov-Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving highorder schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

“…The flux limiter first detects critical numerical fluxes which may lead to negative density and pressure, then limits these fluxes to satisfy a sufficient condition for preserving positivity. Unlike the approaches in [18,19,20], in which positivity-preserving and the maintenance of high order accuracy are considered simultaneously when designing the limiter, here we design the cut-off flux limiter to satisfy positivity only, and then prove a posteriori the maintenance of high order accuracy under a time step restriction. It appears that, in our numerical experiments, a much less restrictive time-step size condition is sufficient for preserving positivity without destroying overall accuracy.…”

“…For even higher-order conservative schemes, Perthame and Shu [11] proved that, given a first-order positivity-preserving scheme, such as Godunov-type schemes, one can always build a higher-order positivity-preserving finite volume scheme under the following constraints: (a) the cell-face values for the numerical flux calculation have positive density and pressure, (b) additional limits on the interpolation under a more restrictive CFL-like condition. With a different interpretation of these constraints based on certain Gauss-Lobatto quadratures, positivity-preserving methods have been successfully developed for high-order discontinuous Galerkin (DG) methods [18] and weighted essentially non-oscillatory (WENO) finite volume and finite difference schemes [19,20].…”

A two-continua approach to Eulerian simulation of water spray

“…Though remedies have been proposed in cases where the maximum principle is satisfied [Zhang and Shu 2011], these do not directly apply in the case of the two-continua equations. During simulation we allow αw / ∈ [0; 1] as practice has shown that αw remains bounded in the discrete case (section 4), and that clamping αw to [0; 1] results in mass conservation issues.…”

On discrete maximum principles for discontinuous Galerkin methods

“…When it comes to discontinuous methods, most of the shock capturing techniques are based on the concept of slope limiter, proposed by Cockburn and Shu for conservation laws [11,10] and latter adapted to the convection-dominated convection-diffusion problem [12]. The same strategy can be applied to finite volume methods (see [33,34,35]). Again, these methods consist in a postprocess after the solution is computed and are designed for explicit methods.…”