A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces

Conventional high-order discontinuous Galerkin (DG) schemes suffer from interface errors caused by the misalignment between straight-sided elements and curved material interfaces. We have developed a novel DG scheme to reduce those errors. Our new scheme uses the correct normal vectors to the curved interfaces, whereas the conventional scheme uses the normal vectors to the element edge. We modify the numerical fluxes to account for the curved interface. Our numerical modeling examples demonstrate that our new discontinuous Galerkin scheme gives errors with much smaller magnitudes compared with the conventional scheme, although both schemes have second-order convergence. Moreover, our method significantly suppresses the spurious diffractions seen in the results obtained using the conventional scheme. The computational cost of our scheme is similar to that of the conventional scheme. The new DG scheme we developed is, thus, particularly useful for large-scale scalar-wave modeling involving complex subsurface structures

A monotone Q1Q^1 finite element method for anisotropic elliptic equations

We construct a monotone continuous $Q^1$ finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific a mesh constraint

A high order accurate bound-preserving compact finite difference scheme for two-dimensional incompressible flow

For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving (SSP) temporal discretizations, we show that the simple bound-preserving limiter in [Li H., Xie S., Zhang X., SIAM J. Numer. Anal., 56 (2018)]. can enforce the strict bounds of the vorticity, if the velocity field satisfies a discrete divergence free constraint. For reducing oscillations, a modified TVB limiter adapted from [Cockburn B., Shu CW., SIAM J. Numer. Anal., 31 (1994)] is constructed without affecting the bound-preserving property. This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field

On the monotonicity of Q2Q^2 spectral element method for Laplacian on quasi-uniform rectangular meshes

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz's condition for proving monotonicity.Comment: arXiv admin note: substantial text overlap with arXiv:2010.0728

A high order accurate bound-preserving compact finite difference scheme for scalar convection diffusion equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak monotonicity property, which implies that a simple limiter can enforce the bound-preserving property without losing conservation and high order accuracy. Higher order accurate compact finite difference schemes satisfying the weak monotonicity will also be discussed