Article

# Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

NASA Ames Research Center, Moffett Field, California, USA; Tel Aviv University, Tel Aviv, Israel; New York University, New York, USA

Journal of Computational Physics (Impact Factor: 2.14). 03/1985; DOI: 10.1016/0021-9991(85)90183-4 Source: NTRS

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**ABSTRACT:**SUMMARYA semi-implicit finite element scheme and a Newton-like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low-order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi-implicit scheme is derived via a time-lagged linearization of the Jacobian operator, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. Copyright © 2011 John Wiley & Sons, Ltd.International Journal for Numerical Methods in Fluids 05/2012; 69(1):1 - 28. · 1.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms. Parabolic terms, when present, can dominate the time for evaluating the solution, especially when large meshes are used. This stems from the fact that the explicit time-step for parabolic terms is proportional to the square of the mesh size and can become unusually small when the mesh is large. Multigrid-Newton-Krylov methods can help, but usually require a large number of iterations to converge. Super TimeStepping schemes are an interesting alternative, because they permit one to take very large overall time-steps for the parabolic terms while using only a modest number of explicit time-steps. Super TimeStepping schemes of the type used in astrophysics have, so far, been only first-order accurate in time and prone to instabilities. In this paper, we present a Runge-Kutta method that is based on the recursion sequence for Legendre polynomials, called the RKL2 method. RKL2 is a time-explicit method that permits us to treat non-linear parabolic terms robustly and with large, second-order accurate time-steps. An s-stage RKL2 scheme permits us to take a time-step that is ˜s2 times larger than a single explicit, forward Euler time-step for the parabolic operator. This permits an s-fold gain in computational efficiency over explicit time-step sub-cycling. For modest values of 's', the advantage can be substantial. The stability properties of the new schemes are explored and they are shown to be stable and positivity preserving for linear operators. We document the method as it is applied to the anisotropic thermal conduction operator for dilute, magnetized, astrophysical plasmas. Implementation-related details are discussed. The RKL2 Super TimeStepping scheme has been implemented in the RIEMANN code for computational astrophysics. We explain the method for picking an s-stage RKL2 scheme for the parabolic terms and show how it can be integrated with a hyperbolic system solver. The method's simplicity makes it very easy to retrofit the s-stage RKL2 scheme to any problem with a parabolic part when a well-formed spatial discretization is available. Several stringent test problems involving thermal conduction in astrophysical plasmas are presented and the method is shown to perform robustly and efficiently on all of them.Monthly Notices of the Royal Astronomical Society 05/2012; 422(3):2102-2115. · 5.52 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shear-thinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.Journal of Fluid Mechanics 09/2013; 731. · 2.29 Impact Factor

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