A toolkit for numerical simulation of PDEs: I. Fundamentals of generic finite-volume simulation

Abstract

To understand underlying physical phenomena or to design better devices or processes, experts in various application areas within science and engineering often need to solve partial differential equations numerically. These experts have tremendous knowledge about the physical phenomena they study, but often are significantly less knowledgeable about scientific computing. A scientific computing toolkit for generic solution of PDEs would be a great benefit for such workers, requiring them only to specify the physics of their problem, with numerical issues handled by the toolkit. This paper presents just such a toolkit, based on the finite-volume method, and argues that the finite-volume method is a more user-friendly choice for such a toolkit than the finite-element method. The user specifies problem physics by providing code snippets to compute interior and boundary fluxes, source terms, and initial conditions; and by specifying constraints on the solution at the boundaries. The toolkit addresses all of the strictly numerical issues: it recovers high-order accurate solution and gradient data from the control volume averaged solution; enforces boundary conditions; integrates the user-supplied fluxes and source terms; and performs time advance. Examples are given for advection–diffusion, solid mechanics, and compressible flow problems to demonstrate the flexibility of the Advanced Numerical Simulation Library framework.