Multigrid for the One-Dimensional Inviscid Burgers Equation

Abstract

A multigrid method for computing steady inviscid compressible flow is investigated for the one-dimensional scalar case. The discretisation in space is obtained by upwind differencing and has first- or second-order accuracy. Only relaxation schemes that affect the solution locally are examined. To obtain some insight into the convergence behaviour, two-level convergence analysis is carried out for the linear constant-coefficient case. The resulting two-grid convergence rates are compared to the asymptotic convergence rates observed in numerical experiments on the nonlinear one-dimensional inviscid Burgers equation.For a test problem with a smooth steady solution, the observed asymptotic convergence rates agreed within $O(h)$ with the linear two-grid convergence rates. A discontinuous solution displayed slower convergence, due to the shock and the sonic point. Although the resulting convergence rate was still acceptable, it could be improved through local relaxation and regularisation of the shock. In this way, a first-order-accurate solution could be obtained in one F-cycle per grid, using damped Point-Jacobi relaxation and successive grid refinement. Second-order accuracy required about eight cycles, using the Defect Correction technique.