Challenges to the State-of-the-practice of Solution Convergence Verification

摘要

The main goal of solution verification is to assess the convergence of numerical predictions as a function of discretization variables such as element size Ax or time step At. The challenge is to verify that the approximate solutions of the discretized laws-of-conservation or equations-of-motion converge to the solution of the continuous equations. In the case of code verification where the continuous solution of a test problem is known, common practice is to obtain several discrete solutions from successively refined meshes or grids; calculate norms of the solution error; and verify the rate with which discrete solutions converge to the continuous solution. With solution verification, where the continuous solution is unknown, common practice is to obtain several discrete solutions from successively refined meshes or grids; estimate an extrapolation of the continuous solution; verify the rate-of-convergence; and estimate numerical uncertainty bounds. The formalism proposed to verify the convergence of discrete solutions derives from postulating how truncation error behaves in the asymptotic regime of convergence. The contribution of this work is to challenge the commonly accepted view of verification and illustrate a number of difficulties encountered during the analysis of general-purpose codes. Examples are given from the disciplines of computational hydro-dynamics that involve the calculation of smooth or shocked solutions of non-linear, hyperbolic equations such as the 1D Burger equation or 2D Euler equations; and engineering mechanics that involve the calculation of smooth solutions of linear or non-linear, elliptic equations. A non-exhaustive list of topics that warrant further research includes: extending the state-of-the-practice to non-scalar quantities (curves, multiple-dimensional fields); studying the coupling between space and time discretizations; defining a reference mesh for the estimation of solution error; and developing technology to verify adaptive mesh refinement calculations in computational engineering and physics. (Approved for unlimited, public release, LA-UR-06-8078, Unclassified.).