Improving the precision of discrete numerical solutions using the generalized integral transform technique

Abstract

Most spatial discretizations applied by finite difference and volume methods to advective/convective terms introduce significant diffusive and dispersive numerical errors, where the former is often required to improve the numerical stability to the resulting scheme. This paper presents a new approach that employs integral transforms to improve the accuracy of available discrete numerical solutions. The method uses a known and inaccurate numerical solution to filter the original governing equations, solves the resulting problem using integral transforms, and then uses it to correct the original numerical solution. Before doing so, these numerical solutions are rewritten as a series representation, which is done using the same eigenfunction basis employed by the integral transformation. Different test cases, based on the unsteady and one-dimensional wave motion described by the nonlinear viscous Burgers' equation in a finite domain, are selected to evaluate the proposed approach. These test cases involve numerical solutions with different types of errors, and are compared with a highly accurate numerical solution for verification. The integral transform technique, in combination with different numerical filters, showed that inaccurate numerical solutions can in fact be corrected by this methodology, as very satisfactory errors were generally obtained. This is true for smooth enough numerical solutions, i.e., the procedure is not advisable to problems that lead to discontinuous numerical solutions.