A Legendre pseudospectral viscosity method
The author considers the scalar conservation law: \[ \partial_t u+\partial_xf(u)=0, \qquad u(x,0)=u_0(x), \] here \(t>0\), \(x\) lies in an open interval and the flux \(f\) is a regular function. It is known that even for smooth \(u_0\) the solution can develop discontinuities, so the problem possesses in general only weak solutions. To ensure uniqueness one needs supplementary conditions. The author considers a so called entropy inequality. To overcome Gibbs type phenomena he proposes the ``Legendre spectral vanishing viscosity'' method which consists in adding a small viscosity on the high coefficients of the numerical solution. This viscosity is small enough to stabilize the computation, ensuring the convergence of the solution of a finite-dimensional variational problem towards the exact one. Test problems show the efficiency to the numerical method.
Legendre spectral vanishing viscosity method
scientific article; zbMATH DE number 947673