The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

Abstract:

We study the stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients. Time is discretized by explicit multi-level or Runge-Kutta methods of order $\leq 3$ (forward Euler time-differencing is included), and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. We prove that these fully explicit spectral approximations are stable provided their time step, $\Delta t$, is restricted by the CFL-like condition $\Delta t < {\text {Const}} \bullet {N^{ - 2}}$, where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of appropriate ${L^2}$-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. Our result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.