Spectral Viscosity Approximations to Multidimensional Scalar Conservation Laws

We study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. We show that the spectral viscosity, which is sufficiently small to retain the formal spectral accuracy of the underlying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, we prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, independent of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. We also show that the SV solution has a bounded total variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. We obtain an L1 convergence rate of the usual optimal order one-half. 1. THE SPECTRAL VISCOSITY APPROXIMATION We consider scalar conservation laws in several space dimensions d, d > 1, (1.1a) dtu(x,t) + dX'f(u(x,t)) = 0, f(u) = (fi(u),f2(u),... ,fd(u)), subject to initial data u(x, 0) = u0(x) G L^r^O, 2n]), and augmented with the entropy condition (cf. [12, 17]) (1.1b) dtU(u) + dx-F(u)<0, Vt/ convex, F(u)= Í U'(w)f'(w)dw. The following abbreviations are used throughout the paper: d= ír a> = lkr a*'oïk*-«'*••••■*>• We want to solve the 2^-periodic initial value problem, ( 1.la)—(1.lb), by a spectral method. To this end, we approximate the spectral/pseudospectral Received by the editor November 25, 1991. 1991 Mathematics Subject Classification. Primary 35L65, 65M06, 65M12, 65M15.