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this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rate-of-strain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall

. We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain "subcharacteristic" condition be satisfied by the hyperbolic system. We support our conjecture with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov's method is coupled with the backward Euler method. We also apply our ideas to the numerical solution of stiff detonation problems. 1. Introduction. Hyperbolic systems of conservation laws with relaxation ...

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. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a thirdorder central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor [NT]. The scalar scheme is non-oscillatory (and hence -- convergent), in the sense that it does not increase the number of initial extrema (--as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the highresolution content of...

Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well-resolved. We design a second order Runge-Kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit, thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. Key words. Hyperbolic conservation laws with stiff relaxation, shock capturing difference method, Runge-Kutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [42,36], in relaxing gas flow with thermal and chemical nonequilibrium [41,9], in kinetic theory of ra...

. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wave-propagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives high-resolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...

this paper, attention is limited to one-dimensional problems, although the procedure is readily extendable to multiple dimensions. The error estimation procedure is applied to a standard, second-order, finite volume discretization of the quasi-one-dimensional Euler equations. Both isentropic and shocked flows are considered. The chosen functional of interest is the integrated pressure along a variable-area duct. The error estimation procedure, applied on uniform grids, provides superconvergent values of the corrected functional. Results demonstrate that additional improvements in the accuracy of the functional can be achieved by applying the proposed adaptive strategy to an initially uniform grid

Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.

In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit second-order Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluid-body interface is based on a volume-of-fluid approach with a redistribution procedure to maintain conservation while avoiding time step restrictions arising from small cells where the boundary intersects the mesh. The single grid Cartesian mesh integration scheme is coupled to a conservative adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. Examples showing the results of the combined Cartesian grid integration/adaptive mesh refinement algorithm for both two- and three-dimensional flows are presented. (This page intent...