New convergence acceleration techniques in the Joe Code

Motivation and objectives

The Joe code is a 2nd-order accurate (in space), finite volume, unstructured cell-based, implicit RANS (Reynolds-averaged Navier-Stokes equations) solver with flamelet-based chemistry (Pecnik et al. 2010a,b) developed within the Stanford PSAAP Center. This code was created to simulate the flow path inside of scramjet engines and, during the past few years, has served as the platform for numerical simulations on complex multiphysics problems. However, some robustness and computational cost aspects of the original code were not sufficiently fine-tuned, and, therefore, the improvement of these aspects has motivated the present research.

At the beginning of 2011, Joe included the following numerical methods for solving the RANS equations: space integration using a HLLC (Harten, Lax, van Leer contact wave) approximate nonlinear Riemann solver (Toro et al. 1994), 2nd-order extrapolation of the convective terms using a MUSCL scheme (Monotone Upstream-centered Schemes for Conservation Laws), and 2nd-order space integration of the viscous terms. Steadystate time integration could be carried out using a multistage Runge-Kutta scheme or a backward Euler method where the linear system is solved using preconditioned GMRES (Generalized Minimum RESidual) or a BiCGSTAB (BiConjugate Gradient STABilized) method.

Using these numerical methods, and when convergence is reached, the Joe solver results in accurate simulations. However, several issues present robustness challenges and prevent the application of the software to full 3D geometry UQ (Uncertainty Quantification) studies. It is important to note that UQ studies using the complete 3D Hyshot II geometry are an important milestone of the PSAAP center and, for that reason, significant effort has been devoted to increasing the robustness and convergence rates of the solver. In particular, during this past year, the following numerical methods have been added to the baseline Joe solver:

• New numerical discretization of the convective terms: both the AUSM (Advection

Upstream Splitting Method) approximate nonlinear Riemann solver (Liou & Steffen

1993) and a JST (Jameson -Schmidt-Turkel) centered scheme (Jameson et al. 1981).

• A geometric, agglomeration-basedmultigrid scheme implemented and tuned for flows

with shocks.

• New linear solvers: BiCGSTAB with linelet preconditioning and a LU-SGS (LU

Symmetric Gauss Seidel) method.

• New estimation of the local time stepping for viscous and turbulent flows to enable

larger CFL numbers.

The objective of this paper is to briefly introduce the new methods that have been implemented in Joe and to show the numerical improvements (robustness, and speed) that those techniques have produced. The overall goal has been to enable the 3D UQ simulations that the Center needs to pursue during the coming year.

F. Palacios/J.J. Alonso Source:

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