AIAA 2004–0763 Practical Implementation and Improvement of Preconditioning Methods for Explicit Multistage Flow Solvers

نویسندگان

  • Kaveh Hosseini
  • Juan J. Alonso
چکیده

Preconditioning methods can help explicit multistage multigrid flow solvers to achieve fast and accurate convergence for a wide range of Mach numbers including incompress-ible flows. The implementation of preconditioning methods and the corresponding matrix dissipation terms in existing flow solvers is a challenging task if good convergence rates are to be obtained. This task can be made more computationally efficient through the use of entropy variables and their associated transformation matrices. Even once implemented , the true potential of preconditioning methods cannot be achieved without a properly chosen entropy fix, well-tuned dissipation coefficients, and a modified Runge-Kutta multistage scheme adapted to the discretization stencil and artificial dissipation terms of the particular flow solver. In this paper we expose the crucial aspects of the successful implementation of a squared preconditioner which can be used in a large class of existing flow solvers that use explicit, modified Runge-Kutta methods and multigrid for convergence acceleration. Numerical and analytical optimization techniques are used to obtain optimal parameter values and coefficients for these methods. The results of these optimizations are used to explore the strengths and weaknesses of both the analytical and numerical approaches and to establish whether the results from both methods are well correlated. Nomenclature l Arbitrary level index for multistage scheme α l Runge-Kutta multistage coefficient β l Modified Runge-Kutta dissipation coefficient W State vector Fx, Fy Flux vectors in the x and y directions Ax, Ay Flux Jacobians in the x and y directions u x-component of velocity v y-component of velocity ρ Density ρ(.) Spectral radius of a matrix p Static pressure E Total energy (internal plus kinetic) H Total enthalpy q Velocity magnitude c Speed of sound R, L Matrices of right and left eigenvectors T Transformation matrix Λ Diagonal matrix of eigenvalues R1, R100 Residuals after 1 and 100 multigrid cycles RD Residual drop after 100 multigrid cycles ∆t Time step ∆x,∆y Spatial increments in the x and y directions λ Courant-Friedrichs-Lewy or CFL number µ Dissipation coefficient

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تاریخ انتشار 2004