A surprisingly effective algorithm for the simplification of integrals and sums arising in the partial differential equations and numerical methods

Abstract

Conservation laws and dissipated quantities are fundamental in analysis. Here we discuss them in partial differential equations and their discretizations. These quantities are integral expressions or, in the discrete case, sums. Often, checking their conservation or dissipation involves tedious computations. Here, we describe a simple algorithm that relies only on linear algebra ideas to simplify these expressions or sums. Also, we discuss another technique to further simplify integral expressions. The, so called, symmetrization that helps to uncover the sign of such integrals. Next, we outline the implementation, including the handling of parametric expressions. Finally, we present some applications to partial differential equations and numerical methods.