An Efficient Formulation of the Modified Nodal Integral Method and Application to the Two-Dimensional Burgers’ Equation

Abstract

An alternate formulation of the recently proposed modified nodal integral method (MNIM) has been developed to further reduce computation time when solving nonlinear partial differential equations with a nonlinear convection term such as Burgers’ equation and the Navier-Stokes equation. In this formulation, by adding and subtracting a linearized convection term, in which the node-averaged velocity at the previous time step multiplies the spatial derivative, the node-interior approximate analytical solution is developed in terms of this previous time-step node-averaged velocity. This leads to a set of discrete equations with coefficients that need to be evaluated only once each time step for each node, resulting in a significant reduction in computing time when compared with the original MNIM formulation. A numerical scheme using the node-averaged velocities at the previous time step—to be referred to as M2NIM—for the two-dimensional, time-dependent Burgers’ equation has been developed. The method is shown to be second order and to posses inherent upwinding. When compared with MNIM, numerical results show a significant reduction in the computation time without sacrificing accuracy.