Accuracy of the Simultaneous-Approximation-Term Boundary Condition for Time-Dependent Problems

The simultaneous-approximation-term (SAT) approach to applying boundary conditions for the compressible Navier-Stokes equations is analyzed with respect to the errors associated with the formulation's weak enforcement of the boundary data. Three numerical examples are presented which illustrate the relationship between the penalty parameters and the accuracy; two examples are fundamentally acoustic and the third is viscous. The viscous problem is further analyzed by a continuous model whose solution is known analytically and which approximates the discrete problem. From the analysis it is found that at early times an overshoot in the boundary values relative to the boundary data can be expected for all values of the penalty parameters but whose amplitude reduces with the inverse of the parameter. Likewise, the long-time behavior exhibits a t (-1/2) relaxation towards the specified data, but with a very small amplitude. Based on these data it is evident that large values of the penalty parameters are not required for accuracies comparable to those obtained by a more traditional characteristics-based method. It is further found that for curvilinear boundaries the SAT approach is superior to the locally one-dimensional inviscid characteristic approach.