Stable integrators for fluid dynamics simulation

Abstract

The partial differential equations of fluid dynamics are discussed in general, and then it is shown how the method of lines is used to discretize these equations to get a large system of ordinary differential equations appropriate for numerical solution techniques. The waterhammer problem is introduced to provide a linear test equation for comparing various numerical methods for solving the differential equation system. The exact solution of the resulting discretized system of equations is developed. The concept of stability of numerical solutions of ordinary differential equations is defined, and the stability properties of several numerical formulas thought to be appropriate for the fluid dynamics problem are presented. The linear test equation is then solved using each of these methods, and the applicability of each is analyzed with respect to its stability characteristics when applied to the waterhammer problem. It is concluded that the trapezoidal formula is most appropriate for transient analysis, but that the two step BDF method is better for steady state and general analysis, at a higher cost in storage.

Keywords:

• Hyperbolic partial differential equations
• parabolic partial differential equations
• method of lines
• ordinary differential equation integrators
• numerical stability
• A-stability
• loss of coolant accident

C.R. Categories:

• 5.17
• 3.20
• 8.1