Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations

摘要

In this paper, we develop parametrized positivity satisfying flux limitersfor the high order finite difference Runge-Kutta weighted essentiallynon-oscillatory (WENO) scheme solving compressible Euler equations to maintainpositive density and pressure. Negative density and pressure, which often leadsto simulation blow-ups or nonphysical solutions, emerges from many highresolution computations in some extreme cases. The methodology we propose inthis paper is a nontrivial generalization of the parametrized maximum principlepreserving flux limiters for high order finite difference schemes solvingscalar hyperbolic conservation laws [22, 10, 20]. To preserve the maximumprinciple, the high order flux is limited towards a first order monotone flux,where the limiting procedures are designed by decoupling linear maximumprinciple constraints. High order schemes with such flux limiters are shown topreserve the high order accuracy via local truncation error analysis and byextensive numerical experiments with mild CFL constraints. The parametrizedflux limiting approach is generalized to the Euler system to preserve thepositivity of density and pressure of numerical solutions via decoupling somenonlinear constraints. Compared with existing high order positivity preservingapproaches [24, 26, 25], our proposed algorithm is positivity preserving by thedesign; it is computationally efficient and maintains high order spatial andtemporal accuracy in our extensive numerical tests. Numerical tests areperformed to demonstrate the efficiency and effectiveness of the proposed newalgorithm.