An efficient formulation of the modified nodal integral method and application to the two-dimensional Burgers' equation

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 M-2 NIM-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.