Comparative Study of the Condition for Non-Oscillatory Solution of a Singularly Perturbed Problem on Uniform and Piecewise-Uniform Meshes
Keywords:uniform mesh, piecewise-uniform mesh, Shishkin mesh, singularly perturbed problem, finite difference method
Wide range of mesh types are proposed in computational fluid dynamics which in turn initiate further discussions over problems of their structure in the numerical computation of fluid flows. Nevertheless, such discussions might sometimes lead to ill-fitted choices of mesh for specific problem. Some types, if improperly used, can cause spurious oscillation in the solutions of governing equations. Furthermore, the contribution of mesh and flow parameters in predicting spurious oscillation free solutions has been much-debated topic over the last decades. Comparison was made in this research between uniform and piecewise-uniform meshes in accentuating the significance of the mesh structure and singular perturbation parameter connection in numerical solution of a singularly perturbed problem. A systematic technique was particularly applied in setting both the singular perturbation parameter and mesh number. Based on the a priori formulation, the condition to avoid spurious oscillatory solutions on the two types of mesh which depends on the parameters of interest is presented in this paper. This was done by adopting reasonable mesh interval sizes. The results of the test cases affirmed the consistency of the condition. It becomes clear that, in general both parameters of interest are linearly related in each case, and the piecewise-uniform mesh number is doubled that of the uniform mesh in order to obtain realistic solution.