In this note,we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations.The idea of this method is to combine a”full-wave”Riemann solver and a”less-wave”Riemann solver,which uses a special modified weight based on the difference in velocity vectors.It is also found that such blending does not need to be implemented in all equations of the Euler system.We point out that the proposed method is easily extended to other”full-wave”fluxes that suffer from shock instability.Some benchmark problems are presented to validate the proposed method.
We propose a new way of rewriting the two dimensional Euler equations and derive an original canonical characteristic relation based on the characteristic theory of hyperbolic systems.This relation contains the derivatives strictly along the bicharacteristic directions,and can be viewed as the 2D extension of the characteristic relation in 1D case.
In this paper,the extremum of second-order directional derivatives,i.e.the gradient of first-order derivatives is discussed.Given second-order directional derivatives in three nonparallel directions,or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions,the formulae for the extremum of second-order directional derivatives are derived,and the directions corresponding to maximum and minimum are perpendicular to each other.
This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions.Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method.Numerical results demonstrate the good performance of the spectral method.
