Pseudo-characteristic method of lines solution of first-order hyperbolic equation systems

Abstract

First-order hyperbolic partial differential equations are difficult to solve numerically because of their ability to transmit steep waves. It is well known that the method of characteristics is the natural method for such equations, as it precisely follows wave interactions. However, a characteristic solution is expensive, as it requires repeated solution of non-linear algebraic equations. This gives considerable motivation to the development of fixed grid numerical schemes. Unfortunately any attempt to use a finite fixed grid generates spurious numerical oscillation and dispersion, which must be minimized by artificial damping or directional differentiation. For sets of hyperbolic equations, the appropriate assignment of damping or direction is difficult to determine, as variables are coupled in non-linear form. However, a clear definition of directionality is given in the characteristic form of the equations, and may be used to develop a pseudo characteristic fixed grid statement of the equations, which is readily solved by the method of lines, is simple to implement, and produces stable accurate solutions. Applications are illustrated for the solution of equations describing shallow water flow, and compressible gaseous flow.