This study concerns the application of a class of `time-iterative' methods to several different classes of problems in fluid mechanics. These new iterative methods combine ideas from multi-stage Runge–Kutta (RK) integration together with a selection of Chebyshev iteration parameters. The underlying performance of these Chebyshev parameterized Runge–Kutta (CPRK) solvers is studied for steady-state solution to a representative sampling from: (i) transport involving convection and diffusion; (ii) incompressible viscous flow governed by the Navier–Stokes equations; (iii) supercritical flume flow in shallow-water theory and (iv) compressible, viscous hypersonic gas flow with multiple reacting species. We provide numerical results to demonstrate convergence to the steady-state flow in each case and give graphs showing residual decay for the CPRK and standard RK schemes.

Timestepping to a steady-state solution is increasingly applied in engineering and scientific applications as a means for solving equilibrium problems. In the present work we examine the relation between the recursion in timestepping algorithms for semidiscrete systems of ODEs and certain types of iterative methods for solving discretized systems of equilibrium PDEs. We consider, in particular, the possibility of accelerating the ODE approach using recursions that are not time accurate together with parameter selection based on the theory of iterative methods. As one example, we take the parameters arising from the Chebyshev-type iterative methods and use them in a two-stage Runge–Kutta scheme. A comparison study for a representative steady-state diffusion problem indicates a dramatic improvement in convergence and efficiency. We remark that this approach can be trivially incorporated into existing time-integration codes to significant advantage. This yields a hybrid adaptive approach in a single code.

A class of vector-parallel schemes for solution of steady compressible or incompressible viscous flow is developed and performance studies carried out. The algorithms employ an artificial transient treatment that permits rapid integration to a steady state. In the present work a four-stage explicit Runge-Kutta scheme employing variable local step size is utilized for the ODE system integration. The RK-4 scheme is restructured to allow vectorization and enhance concurrency in the calculation for a streamfunction-vorticity formulation of the flow problem. The parameters of the resulting RK scheme can be selected to accelerate convergence of the KK recursion. Four main procedures are considered which permit vector-parallel solution: a Jacobi update, a hybrid of the Jacobi and Gauss-Seidel method, red-black ordering and domain decomposition. Numerical performance studies are conducted with a representative viscous incompressible flow calculation. Results indicate that a scheme involving domain decomposition with a Gauss-Seidel type of update for the RK four-stage scheme is most effective and provides performance in excess of 8 Gflops on the Gray C-90.

Time-iterative strategies for obtaining steady state solutions to the shallow water equations are developed using reduced-order, multistage Runge-Kutta (RK) methods. The shallow water system is formulated using a streamline upwind/Petrov-Galerkin technique that includes an additional operator for control of steep gradients. Numerical results are given to illustrate the performance of the RK methods.