Spectral Viscosity

We study the behavior of spectral viscosity approximations to nonlinear scalar conservation laws. We show how the spectral viscosity method compromises between the total-variation bounded viscosity approximations { which are restricted to rst-order accuracy, and the spectrally accurate { yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is L 1-stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be Lip +-stable in agreement with Oleinik's E-entropy condition. This essentially non-oscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution and we provide convergence rate estimates of both-global and local types. 1. THE SPECTRAL VISCOSITY APPROXIMATION We are concerned here with spectral approximations of the scalar conservation law (1:1a) @ @t u(x; t) + @ @x f(u(x; t)) = 0; u(x; 0) u 0 (x)BV: To single out a unique physically relevant weak solution, (1.1a) is complemented with an entropy condition such that for all convex U's, (e.g., La],,Sm]) (1:1b) @ @t U(u) + @ @x F(u) 0; F(u) Z u U 0 ()f 0 ()dd: We want to solve the 2-periodic initial-value problem (1.1a)-(1.1b) by spectral methods. To this end we use an N-trigonometric polynomial, u N (x; t) = P N k=?N ^ u k (t)e ikx , to approximate the spectral (or pseudospectral) projection of the exact entropy solution, P N u. (1:2) @ @t u N + @ @x P N f(u N) = 0: Together with one's favorite ODE solver, (1.2) gives a fully discrete spectral method for the approximate solution of (1.1a). Although the spectral method (1.2) is a spectrally accurate approximation of the conservation law (1.1a), in the sense that its local error does not exceed (1:3) the spectral solution, u N (x; t), need not approximate the corresponding entropy solution, u(x; t). Indeed , counterexamples provided in Ta1],,Ta2] show that the spectral approximation (1.2) lacks entropy dissipation, which is inconsistent with the entropy condition (1.1b). Consequently, the spectral approximation (1.2) supports spurious Gibbs oscillations which prevent strong convergence to the exact solution of (1.1). To suppress these oscillations, without sacriicing the overall spectral accuracy, we consider instead the Spectral Viscosity (SV) approximation (1:4) @ @t u N (x; t) + @ @x P N f(u N (x; t)) = " N @ @x Q N @ @x u N (x; t):