Semi-Lagrangian Solutions to the Inviscid Burgers Equation

Abstract

We explore the use of semi-Lagrangian methods in a situation where the spatial scale of the flow collapses to zero during the time integration. The inviscid Burgers equation is used as the test model because it is the simplest equation that allows scale collapse (shock formation), and because it has analytic solutions. It is shown that despite the variable manner in which the gradient of the wind field approaches infinity in the neighborhood of the shock, the semi-Lagrangian method allows the error to be localized near the steep slope region. Comparisons with second-order finite difference and tau methods are provided. Moreover, the semi-Lagrangian method gives accurate results even with larger time steps (Courant number greater than 2 or 4) than are possible with the Eulerian methods. The semi-Lagrangian method, along with other recently developed numerical methods, is useful in simulating the development Of steep gradients or near discontinuities in a numerical model. Some applications of the semi-Lagrangian method are discussed.