A Brief Overview of Code and Solution Verification in Numerical Simulation

This manuscript is a brief overview of code and calculation verification in computational physics and engineering. Verification is an essential technology to assess the quality of discrete solutions obtained by running simulation codes that solve systems of ordinary or partial differential equations, such as the finite element and finite volume methods. Code verification assesses the extent to which a numerical method is implemented correctly, that is, without any programming mistake (or “bug”) that would adversely affect the quality of computations. The centerpiece of code verification is the formulation of verification test problems that admit exact or manufactured solutions and which are used for comparison with approximate solutions obtained from the simulation software. Solution verification assesses the extent to which the discretization (in time, space, energy, modal basis, etc.) implemented to solve governing equations provides a sufficiently small level of truncation error. The keystone of solution verification is the practice of mesh refinement from which estimates of the (spatial) order of accuracy of the numerical method can be estimated. It is also possible to derive bounds of truncation error produced in the calculation. The discussion is presented in the context of Peter Lax’s 1954 groundbreaking work on the convergence of discrete solutions. It is illustrated with a simple example of one-dimensional advection solver. (Publication approved for unlimited, public release, LA-UR-16-24553, Unclassified.)