On the multi-level splitting of finite element spaces
Numerische Mathematik
Hierarchical bases give conjugate gradient type methods a multigrid speed of convergence
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
Analysis of the spectral vanishing viscosity method for periodic conservation laws
SIAM Journal on Numerical Analysis
Convergence of spectral methods for nonlinear conservation laws
SIAM Journal on Numerical Analysis
Legendre pseudospectral viscosity method for nonlinear conservation laws
SIAM Journal on Numerical Analysis
ICIAM 91 Proceedings of the second international conference on Industrial and applied mathematics
Introduction to scientific computing: a matrix-vector approach using MATLAB
Introduction to scientific computing: a matrix-vector approach using MATLAB
Enhanced spectral viscosity approximations for conservation laws
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Spectral Vanishing Viscosity Method For Nonlinear Conservation Laws
SIAM Journal on Numerical Analysis
Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws
SIAM Journal on Numerical Analysis
Second-order Godunov-type scheme for reactive flow calculations on moving meshes
Journal of Computational Physics
A Finite Element, Multiresolution Viscosity Method for Hyperbolic Conservation Laws
SIAM Journal on Numerical Analysis
An adaptive wavelet viscosity method for systems of hyperbolic conservation laws
Journal of Computational and Applied Mathematics
Hi-index | 31.45 |
It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flows. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution developed in the context of spectral methods by Eitan Tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy. We incorporate this idea into the finite element framework by using hierarchical functions as a multi-frequency basis. The result is a new finite element method for solving hyperbolic conservation laws. For this method, convergence for a one-dimensional scalar conservation law has previously been proved. Here, the method is described in detail, several issues connected with its efficient implementation are considered, and numerical results for several examples involving one- and two-dimensional hyperbolic conservation laws are provided. Several advantageous features of the method are discussed, including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space.