Calculation Verification: Pointwise Estimation of Solutions and Their Method-Associated Numerical Error

Abstract

Calculation verification is a method of quantifying the convergence properties of a specific implementation of an algorithm without using an exact solution, whereas code verification does use an exact solution. Calculating global convergence while performing calculation verification can be problematic as no exact solution is generally available. A second difficulty sometimes encountered in both code and calculation verification is when the computed solutionsatapointinspaceconvergeinanoscillatorymanner,makingcalculationofconvergence difficult.An extension to the standard method of calculation and code verification is explored which may address both of these issues. The proposed method uses pointwise convergence analysis to estimate global convergence behavior by first estimating a discrete solution to the partial differential equations. The estimated solution is then used in place of an exact solution to evaluate global convergence rates when conducting a calculation verification analysis. Additionally, the absolute value of the pointwise error is calculated, allowing for monotonic or oscillatory convergence. This method was tested on four pure hydrodynamics problems usingtheEuleriancodeRAGE.Alinearacousticwaveanda1-DRiemannproblembothhave exact mathematical solutions, which were compared to the estimated solution. For these two problems both code and calculation verification methods were applied.A nonlinear acoustic wave and 2-D Riemann problem both have no exact solutions, but the estimated solution was shown to be useful in providing a more accurate solution on a coarse grid than the calculated solution. For these two problems, only a calculation verification analysis was conducted.