Maximum-principle-satisfying And Positivity-preserving High Order Central Dg Methods On Unstructured Overlapping Meshes For Two Dimensional Conservation Laws

In this paper,central discontinuous Galerkin methods defined on unstructured overlapping meshes for two dimensional conservation laws are reserched.As a family of high order numerical methods,central discontinuous Galerkin(CDG)methods possess the characters of discontinuous Galerkin(DG)methods and central methods.Because of the discontinuity of base functions of CDG methods,CDG methods have advanteges of high order,highly parallelizability,and easily to consturct the high order schemes.Moreover,combining the ideas of central methods,CDG methods is defined on two overlapping meshes,which brings better stability of the methods.And,CDG methods are free of Rimann functions on each elements interface.However,there are difficulties in numerical calculation and error analysis in theory,for CDG methods need to calculate numerical soluitons on two meshes.First of all,most of works relating to CDG methods are defined on structured meshes.While the CDG methods on structured meshes is only suitable for solving problems on simple regions.Therefore,an approach of construction of unstructured overlapping meshes is presented in this paper.And,the high order CDG methods is defined on the defined unstructured overlapping meshes for solving two dimensional conservation laws.For solving linear conservation laws,the L2 stability of CDG methods on unstructured overlapping laws is proved.The performance of the proposed methods is finally demonstrated through a set of numerical experiments.Different problems on complex regions are solved for showing the advantege of unstructured meshes.Secondly,The unique entropy solution to scaler conservation laws satisfies a strict maximum principle,that is,the solution preserves between the maximum and mininum values.Therefore,numerical methods for solving scaler conservation laws should keep the numerical soluiton satisfing the maximum principle.Based on the ideas in maximum-principle-satisfing DG methods,the CDG methods satisfing maximum principle on unstructured overlapping meshes for solving scaler conservation laws are presented.To design the maximum-principle-satisfing mathods,first we present the principle of maximum presering CDG methods for the mean value of numerical solution on each elements,setting the mean value of soluiton on each elements to be maximum presering with the CFL condition satisfied.Then a maximum-principle-satisfing limiter is designed,which modifies the solution to be maximum presering using the mean value.Thirdly,the density and pressure in Euler equations need to hold positive during numerical calculation.Therefore,it is important for solving Euler equations to design positivity presering methods.In this paper,the positivity presering CDG methods on unstructured overlapping meshes is designed using ideas similar to maximum presering methods.To design the positivity-presering mathods,first we present the principle of positivity-presering CDG methods for the mean value of numerical solution on each elements,setting the mean value of soluiton on each elements to be positivity-presering with the CFL conditions satisfied.Then a positivity-presering limiter is designed,which modifies the solution to be positivity-presering using the mean value.The validity and error demonstration of the proposed methods are accomplished through comparing the numerical soluitons with and without the corresponding limiters.