Analysis of Computational Methods for Nonlinear Parabolic Differential Systems

摘要

Many important problems in fluid dynamics, among other areas, are modeled by nonlinear parabolic differential systems with initial values given in one 'independent variable' x, and boundary values in the remaining dependent variables. Hyperbolic systems can sometimes be treated as a special case. For example, the inviscid flow case of the Navier-Stokes equations is a hyperbolic system, while the viscous flow case is elliptical. A survey of currently used numerical methods is in Richtmeyer and Morton. In subsonic flow cases, the nonlinear terms are small enough to be ignored, but these terms must be included in supersonic and hypersonic flow. These numerical calculations usually involve a finite difference mesh over the boundary value problem variables, resulting in a space discretization matrix equation which for the nonlinear system varies at each step in x, the independent variable representing time in the dynamic case or one of the space variable for the steady state case. Then this nonlinear system is solved as an initial value problem in x. The initial value problem is usually solved by a one step implicit method for reasons of cost and stability. Some methods based on finite element methods for the boundary value problem can be used, but successful methods are only available for the linear cases, such as subsonic flow problems.