Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.
In this paper, we develop, analyze and test the Fourier spectral
methods for solving the Degasperis-Procesi(DP) equation which contains nonlinear
high order derivatives, and possibly discontinuous or sharp transition solutions.
The $L^2$ stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method.By applying the Gegenbauer reconstruction technique as a post-processing method to the
Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully.
The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided
to illustrate the accuracy and capability of the methods.
In this paper, we develop the high order weighted essentially non-oscillatory (WENO) schemes for solving the Degasperis-Procesi (DP) equation, including finite volume (FV) and finite difference (FD) methods. The DP equation contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The finite volume method is de- signed based on the total variation bounded property of the DP equation. And the finite difference method is constructed based on the L2 stability of the DP equation. Due to the adoption of the WENO reconstruction, both schemes are arbitrary high order accuracy and shock capturing. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.
In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, some pairs of $H^1$-conforming finite element spaces such as $CP_1$-$P_1$, $CP_2$-$P_1$, and $P_2$-$P_1$ elements for electric field and multiplier are studied, and their stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed
mixed $H^1$-conforming finite element method.