Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

Abstract

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8,9,19,21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0 , β 1 ) in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter. and practically cause the computations to blow up. Thus it is attractive to have the numerical solution satisfy discrete maximum principle (or preserve positivity). Solution of equation (1.1) may represent a specific physical meaning and is supposed to be positive, thus negative value approximation loses physical meanings in such cases.Generally it is very difficult to design high order numerical methods that satisfy discrete maximum principle for convection diffusion equations (1.1). No finite difference method is known to achieve better than second-order accuracy [5,25,20] that satisfies discrete maximum principle. Much less is known for higher order methods such as spectral FEM, hp-FEM or finite volume methods [2,6,13,23]. Compared to the elliptic type, more restrictive conditions on mesh are required to obtain discrete maximum principle for the parabolic type equations, see [18,16,7,17].In this article, we study direct discontinuous Galerkin method [8] and its variations [9,19,21], and prove the polynomial solution satisfy discrete maximum principle with third order of accuracy. Discontinuous Galerkin (DG) method is a class of finite element method that use completely discontinuous piecewise functions as numerical approximations. Since the basis functions can be completely discontinuous, these methods have the flexibility that is not shared by standard finite element methods, such as the allowance of arbitrary triangulations with hanging nodes, complete freedom of choosing polynomial degrees in each element (p adaptivity), and extremely local data structure and the resulting high parallel efficiency.Recently in [24,25,26], Zhang and Shu designed a maximum-principle-satisfying limiter for high order DG and finite volume methods for hyperbolic conservation laws. The key step in Zhang and Shu's discussion is to show the polynomial solution average falling in the given minimum and maximum bounds. For hyperbolic type equations, the solution average evolution only relies on the solution polynomial values on the element edges. For diffusion type equations, the evolution of solution average depends on the solution derivative values on the edges, thus the technique developed in [24] can not be applied.In [8], we developed the direct DG method (DDG) as a new diffusion solver. The key contrib...

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“…Unfortunately, weak monotonicity holds only up to second order accuracy for any linear finite volume scheme and most DG schemes [24], see Appendix D. Surprisingly, it is still possible to construct a third order linear DG scheme satisfying the weak monotonicity. With special parameters, the direct DG (DDG) method, which is a generalized version of interior penalty DG method, indeed satisfies a weak monotonicity up to third order accuracy [25,26,27,28]. However, if we use Taylor expansion to examine the local truncation error in the numerical flux of this scheme, only second order accuracy is obtained.…”

An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

“…The maximum‐principle for (4.22) means that if , then for all t > 0, which implies the usual maximum‐principle for diffusion for all t > 0. Extension to unstructured meshes is nontrivial, we refer to for some recent results in solving diffusion equations over triangular meshes.…”

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

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