On the Conservation and Convergence to Weak Solutions of Global Schemes
Journal of Scientific Computing
A multidomain spectral method for supersonic reactive flows
Journal of Computational Physics
A spectral viscosity method for correcting the long-term behavior of POD models
Journal of Computational Physics
Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor-Green Vortex Flow
Journal of Scientific Computing
The Chebyshev-Legendre collocation method for a class of optimal control problems
International Journal of Computer Mathematics
Preconditioning on high-order element methods using Chebyshev--Gauss--Lobatto nodes
Applied Numerical Mathematics
Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations
Journal of Scientific Computing
A Legendre Petrov-Galerkin method for fourth-order differential equations
Computers & Mathematics with Applications
Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions
SIAM Journal on Scientific Computing
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In this paper, a Chebyshev--Legendre spectral viscosity (CLSV) method is developed for nonlinear conservation laws with initial and boundary conditions. The boundary conditions are dealt with by a penalty method. The viscosity is put only on the high modes, so accuracy may be recovered by postprocessing the CLSV approximation. It is proved that the bounded solution of the CLSV method converges to the exact scalar entropy solution by compensated compactness arguments. Also, a new spectral viscosity method using the Chebyshev differential operator $D=\sqrt{1-x^2} \opx$ is introduced, which is a little weaker than the usual one while guaranteeing the convergence of the bounded solution of the Chebyshev Galerkin, Chebyshev collocation, or Legendre Galerkin approximation to nonlinear conservation laws. This kind of viscosity is ready to be generalized to a super viscosity version.