Distributed minimal residual (DMR) method for acceleration of iterative algorithms

Abstract

A new method for enhancing the convergence rate of iterative schemes for the numerical integration of systems of partial differential equations has been developed. It is termed the Distributed Minimal Residual (DMR) method, and is based on general Krylov subspace methods. The DMR method differs from the Krylov subspace methods by the fact that the iterative acceleration factors are different from equation to equation in the system. At the same time, the DMR method can be viewed as an incomplete Newton iteration method. The DMR method has been applied to Euler equations of gasdynamics and incompressible Navier-Stokes equations. All numerical test cases were obtained using either explicit four stage Runge-Kutta or Euler implicit time integration. The DMR method was found capable of reducing the computation time by 20–80% depending on the test case. When directly compared with an implicit residual smoothing, the DMR method performed consistently better and more reliably. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.