Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Abstract

SUMMARYSemidirect solution techniques can be an effective alternative to the more conventional iterative approaches used in many finite difference methods. This paper summarizes several semidirect techniques which generally have not been applied to the Navier-Stokes and energy equations in finite difference form. The methods presented use both successive substitution and Jacobian-based updates as well as two variations of Broyden's full matrix update. A hybrid method is also presented, as is a norm-reducing search technique that can be used to enhance the convergence characteristics of Any semidirect approach. These methods have been compared with the well known iterative methods SIMPLE and SIMPLER. The comparison was performed on the natural convection and driven cavity problems. The semidirect methods proved to be reliably convergent without the need for a priori specification of variable under-relaxation factors, which was necessary with the iterative methods. Natural convection and driven cavity solutions have been readily obtained with the proposed methods for Rayleigh and Reynolds numbers up to 10" and 10' respectively. Of the semidirect techniques, the hybrid approach was the most robust. From an arbitrary zero initial guess this method was able to obtain a solution to the natural convection problem for Rayleigh numbers three orders of magnitude larger than was possible with the Newton--Raphson update. The computational effort required by the semidirect methods is comparable to that required by the iterative methods; however, the memory requirements can be significantly greater.

A new finite element formulation for incompressible flow

“…As discussed by MacArthur and Patankar [4], since the segregated solver reduces the residual only locally a t each step of the solution process, its convergence rate is strongly influenced by the number of unknowns.…”

“…However, segregated solvers sometimes cannot adequately handle strong coupling because of the approximations that produce the linearized systems. In addition, since the segregated solution approach reduces the residual only locally at each step, the number of iterations required for convergence and the computational effort tend to increase significantly as grids are refined (see, e.g., [4]). …”

Defect Correction With a Fully Coupled Inexact Newton Method

“…The nonlinear system is then linearized using Newton's method. It has been found that a fully coupled approach used with Newton's method can successfully achieve solutions that are not easily obtained using a sequential solver [9,11,121. An inexact Newton's method [13,141 employs an iterative solution technique to solve the linear system rather than a direct solver.…”

Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

“…Primitive-variable formulations of incompressible Navier-Stokes fluid-flow problems are notoriously difficult to solve numerically. A robust technique is to use Newton iteration for the fully-coupled nonlinear equations [1,2,3,4,5,6,7,8,9,10]. This technique is commonly used at present since it requires few (if any) iteration parameters and converges more quickly than segregated approaches such as SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) [11,12].…”

“…A direct matrix method can be used to solve the Jacobian [1,2,4,6,7,8,9,10] but this is prohibitively expensive for three-dimensional problems. An alternative is to use an iterative matrix method.…”