Compact HigherOrder Numerical Methods for Hyperbolic Conservation Laws
Abstract
A method is developed for the simulation of nonlinear wave propagation over long times. The approach is based on the spacetime Discontinuous Galerkin finiteelement method (DG). In deriving the method, the idea of compactness is strictly followed. That is, that the discrete domain of dependence should contain a minimum amount of data outside the physical domain of dependence. Compactness is attained by carrying higherorder solution moments in each spatial mesh cell, and through the definition of a family of space time meshes. The meshes also eliminate the need for operator splitting in multidimensional problems. For any order of accuracy, the resulting method is stable for Courant numbers less than 1, satisfies an entropy condition, and uses the same boundary procedure. In addition, the method can be written as a minimization of the jump in the numerical solution along the inflow face of each element. A Fourier analysis of the DG(k) method is given, where k is the order of the polynomial used in each element. The method has a 'superconvergence' property, in that the evolution error is {cal O}(h^{2k+1 }). Other norms have slower errorconvergence rates, as a result of the error in the initial projection onto the accurate mode. However, this initial error is often small, and may be overwhelmed by the evolution error for longtime problems. Numerical experiments indicate that the superconvergence property extends to some nonlinear cases. The consequences of mesh staggering in time, in an effort to eliminate the need for the solution of the Riemann problem, are also investigated. The diffusion that results from staggering is usually insignificant for unsteady onedimensional flows. For multidimensional flows, the effects are more pronounced, particularly for meshaligned shear flows. Numerical results are presented for scalar advection and the Euler equations, in both one and twospace dimensions. Comparisons are made with a standard secondorder finitevolume scheme. DG(k) is shown to have high accuracy, but unless a small error tolerance is required, or the evolution time is large, the cost is high. Suggestions are made for reducing the cost, and for developing related methods.
 Publication:

Ph.D. Thesis
 Pub Date:
 1996
 Bibcode:
 1996PhDT........42L
 Keywords:

 COMPUTATIONAL FLUID DYNAMICS;
 DISCONTINUOUS GALERKIN;
 SPACE TIME;
 NONLINEAR;
 WAVE PROPAGATION;
 Engineering: Aerospace; Engineering: Mechanical; Physics: Acoustics