A General Solution Verification Method for Complex Heat and Flow problem with Hands off Coding

This paper addresses the challenge of Solution Verification (SV) and accuracy assessment for computing complex Partial Differential Equation (PDE) model. Our goal is to provide a postprocessing package that can be attached to any existing numerical simulation package, for example, widely used commercial codes such as ADINA, Ansys, Fluent, Numeca, StarCD etc... and provide an a posteriori error estimate to their simulation. Important design decisions are based on simulation done with these softwares. Unfortunately, we know that to verify a numerical solution, that is to provide a quantitative assessment on the numerical accuracy of the solution, is difficult. The problem of accuracy assessment is a necessary step that comes after the code verification step and before the code validation step to complete the global task of providing a reliable virtual experiment tool [8, 9]. Our major goal in this paper is to pursue our work on the design of a new method that offers a general framework to do solution verification efficiently [2, 3, 1]. The standard approach in applied mathematics to handle the problem of solution verification is to work on the approximation theory of the PDE. For each specific PDE problem, the right Finite Element (FE) approximation may provide the correct a posteriori error estimate [10]. Unfortunately, this approach may require a complete rewriting of an existing Computational Fluid Dynamic (CFD) code based on Finite Volume (FV) for example, and lack generality. Usually a posteriori estimators fails if the (nonlinear) PDE solution is stiff or if the grid resolution is not adequate. Since grid refinement itself is based on a posteriori estimator, this poses an obvious problem. Large Reynolds number flow are common in many applications, not to mention turbulence problems. For those applications rigorous solution verification may not be achievable by the current state of the art of numerical analysis. The difficulty of SV is even greater for complex multi-physic coupling. The general practice in scientific computing is to simulate PDEs, for which applied mathematics, neither numerical analysis, guaranty the result.