A novel physics-based preconditioner for nodal integral method using JFNK for 2D Burgers equation

Abstract

Nodal Integral Methods (NIM) are prevalent for solving neutron transport equations. Due to the success of this method for solving neutron transport problems, the method was used for solving fluid flow problems. In NIM, first PDEs are averaged over a node to form the ODEs and final schemes are developed using the analytical solution of these ODEs. Further modifications were made, leading to two versions called modified NIM (MNIM) and modified MNIM (M2NIM). These successive improvements were made to make it more robust for fluid flow problems, still it is limited to low Reynold number (Re) problems. The shortfall of nonlinear solvers compatible with NIM is the main reason for this limitation. Jacobian Free Newton Krylov (JFNK) in combination with NIM, is one way to improve the capabilities of NIM. However, the number of Krylov iterations can be quite large for high Re case. Here, a physics-based preconditioner is developed using linearized M2NIM. The developed algorithm is tested for the simulation of shock wave using 2D Burgers' equation with larger time steps and higher Re and found that the method is very accurate even with a coarse grid. The method is very effective in reducing computational time and Krylov iterations.