Finite Difference Weighted Essentially Non-oscillatory Research Articles

A Technique of Treating Negative Weights in WENO Schemes

High-order accurate weighted essentially nonoscillatory (WENO) schemes have recently been developed for finite difference and finite volume methods both in structured and in unstructured meshes. A key idea in WENO scheme is a linear combination of lower order fluxes or reconstructions to obtain a higher order approximation. The combination coefficients, also called linear weights, are determined by local geometry of the mesh and order of accuracy and may become negative, such as in the central WENO schemes using staggered meshes, high-order finite volume WENO schemes in two space dimensions, and finite difference WENO approximations for second derivatives. WENO procedures cannot be applied directly to obtain a stable scheme if negative linear weights are present. The previous strategy for handling this difficulty is either by regrouping of stencils or by reducing the order of accuracy to get rid of the negative linear weights. In this paper we present a simple and effective technique for handling negative linear weights without a need to get rid of them. Test cases are shown to illustrate the stability and accuracy of this approach.

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics

We present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio–Wu Riemann problems, the two-dimensional Kelvin–Helmholtz instability problems, and the two-dimensional Orszag–Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.

Efficient Implementation of Weighted ENO Schemes

In this paper, we further analyze, test, modify, and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher, and Chan. It was shown by Liuet al.that WENO schemes constructed from therth order (inL1norm) ENO schemes are (r+ 1)th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a fifth-order WENO scheme for the caser= 3, instead of the fourth-order with the original smoothness measurement by Liuet al.This fifth-order WENO scheme is as fast as the fourth-order WENO scheme of Liuet al.and both schemes are about twice as fast as the fourth-order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest computing the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights and work well for problems which do not contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy wave interaction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.

Third order WENO scheme on sparse grids for hyperbolic equations

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). The computational cost of such schemes increases significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points and nonlinearity of high order accuracy WENO schemes. How to achieve fast computations by WENO methods for high spatial dimension PDEs is a challenging and important question. Recently, sparse-grid has become a major approximation tool for high dimensional problems. The open question is how to design WENO computations on sparse grids such that comparable high order accuracy of WENO schemes in smooth regions and essentially non-oscillatory stability in non-smooth regions of the solutions can still be achieved as that for computations on regular single grids? In this paper, we combine the third order finite difference WENO method with sparse-grid combination technique and solve high spatial dimension hyperbolic equations on sparse grids. WENO interpolation is proposed for the prolongation part in sparse grid combination techniques to deal with discontinuous solutions of hyperbolic equations. Numerical examples are presented to show that significant computational times are saved while both high order accuracy and stability of the WENO scheme are maintained for simulations on sparse grids.