High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

An improved third-order HWCNS for compressible flow simulation on curvilinear grids

Due to the very high requirements on the quality of computational grids,stability property and computational efficiency,the application of high-order schemes to complex flow simulation is greatly constrained.In order to solve these problems,the third-order hybrid cell-edge and cell-node weighted compact nonlinear scheme(HWCNS3)is improved by introducing a new nonlinear weighting mechanism.The new scheme uses only the central stencil to reconstruct the cell boundary value,which makes the convergence of the scheme more stable.The application of the scheme to Euler equations on curvilinear grids is also discussed.Numerical results show that the new HWCNS3 achieves the expected order in smooth regions,captures discontinuities sharply without obvious oscillation,has higher resolution than the original one and preserves freestream and vortex on curvilinear grids.

A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO(SWENO)scheme.The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism.The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENOscheme efficiently.Besides,a scaling technique is used to circument the growth of condition numbers as mesh refined.However,weak oscillations still appear when the SWENO scheme deals with complex low density equations.In order to guarantee the maximum-principle-preserving(MPP)property,we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy.A novel procedure is designed to prove this property theoretically.Finally,numerical examples for one-and two-dimensional problems are presented to verify the good performance,maximum principle preserving,essentially non-oscillation and high resolution of the proposed scheme.

A Well-Balanced Weighted Compact Nonlinear Scheme for Pre-Balanced Shallow Water Equations

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.