Total Variation and Error Estimates for Spectral Viscosity Approximations

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 first-order accuracy—and the spectrally accurate, yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is Ll-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 nonoscillatory 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) — u(x,t) + — fiuix, t)) = 0, u(x,P) = u0(x)£BV. To single out a unique physically relevant weak solution, ( 1. la) is complemented with an entropy condition such that for all convex U 's (e.g., [7, 12]) (Lib) ^U(u) + -^F(u) < 0, F(u) = j" U'it)fii)de:. We want to solve the 2^-periodic initial value problem (1.1 a)-( 1. lb) by spectral methods. To this end, we use an TV-trigonometric polynomial, u^ix, t) = J2k=-Nuk(t)e'kx > to approximate the spectral (or pseudospectral) projection of the exact entropy solution, PNu . Starting with Un(x , 0) = PNu0(x), the standard Fourier method reads, e.g., [5, 2, 1], (1.2) -uN + —PNf(uN) = P. Together with one's favorite ODE solver, (1.2) gives a fully discrete spectral method for the approximate solution of (1.1a). Received by the editor April 9, 1991 and, in revised form, August 16, 1991. 1991 Mathematics Subject Classification. Primary 35L65, 65M06, 65M12, 65M15.