Methods for Multi-Dimensional PDEs

Abstract

This chapter is concerned with numerical methods for solving non-linear systems of hyperbolic conservation laws in multidimensions. For Cartesian geometries one may write the equations of our interest here as

$${U_t} + F{\left( U \right)_x} + G{\left( U \right)_y} + H{\left( U \right)_z} = 0,$$

(16.1)

where t denotes time or a time-like variable and x, y, z are Cartesian coordinate directions. U is the vector of conserved variables and F(U), G(U), H(U) are vectors of fluxes in the x, y, z directions respectively. A prominent example are the time-dependent three dimensional Euler equations studied in Sect. 3.2 of Chap. 3. Other examples are the time-dependent two dimensional shallow water equations and the artificial compressibility equations. See Sect. 1.6.3 of Chap. 1.