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NASA NTRS1985

A nonoscillatory shock capturing scheme using flux limited dissipation

A method for modifying the third order dissipative terms by the introduction of flux limiters is proposed. The first order dissipative terms can then be eliminated entirely, and in the case of a scalar conservation law the scheme is converted into a total variation diminishing scheme provided that an appropriate value is chosen for the dissipative coefficient. Particular attention is given to: (1) the treatment of the scalar conservation law; (2) the treatment of the Euler equations for inviscid compressible flow; (3) the boundary conditions; and (4) multistage time stepping and multigrid schemes. Numerical results for transonic flows suggest that a central difference scheme augmented by flux limited dissipative terms can lead to an effective nonoscillatory shock capturing method.

Jameson, A.↗
NASA NTRS1985

Application of TVD schemes for the Euler equations of gas dynamics

Highly accurate and yet stable shock-capturing finite difference schemes have been designed for the computation of the Euler equations of gas dynamics. Four different principles for the construction of high resolution total variation diminishing (TVD) schemes are available, including hybrid schemes, a second-order extension of Godunov's scheme by van Leer (1979), the modified flux approach of Harten (1983, 1984), and the numerical fluctuation approach of Roe (1985). The present paper has the objective to review the class of second-order TVD schemes via the modified flux approach. Attention is given to first-order TVD schemes, a second-order accurate explicit TVD scheme, the global order of accuracy of the second-order TVD scheme, extensions to systems and two-dimensional conservation laws, numerical experiments with a second-order explicit TVD scheme, implicit TVD schemes, and second-order implicit TVD schemes.

Yee, H. C.↗
NASA NTRS1985

On the implementation of a class of upwind schemes for system of hyperbolic conservation laws

The relative computational effort among the spatially five point numerical flux functions of Harten, van Leer, and Osher and Chakravarthy is explored. These three methods typify the design principles most often used in constructing higher than first order upwind total variation diminishing (TVD) schemes. For the scalar case the difference in operation count between any two algorithms may be very small and yet the operation count for their system counterparts might be vastly different. The situation occurs even though one starts with two different yet equivalent representations for the scalar case.

Yee, H. C.↗
NASA NTRS1985

On symmetric and upwind TVD schemes

A class of explicit and implicit total variation diminishing (TVD) schemes for the compressible Euler and Navier-Stokes equations was developed. They do not generate spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1 to 2 grid points. For the inviscid case, these schemes are divided into upwind TVD schemes and symmetric (nonupwind) TVD schemes. The upwind TVD scheme is based on the second-order TVD scheme. The symmetric TVD scheme is a generalization of Roe's and Davis' TVD Lax-Wendroff scheme. The performance of these schemes on some viscous and inviscid airfoil steady-state calculations is investigated. The symmetric and upwind TVD schemes are compared.

Yee, H. C.↗
NASA NTRS1986

Advances in numerical and applied mathematics

This collection of papers covers some recent developments in numerical analysis and computational fluid dynamics. Some of these studies are of a fundamental nature. They address basic issues such as intermediate boundary conditions for approximate factorization schemes, existence and uniqueness of steady states for time dependent problems, and pitfalls of implicit time stepping. The other studies deal with modern numerical methods such as total variation diminishing schemes, higher order variants of vortex and particle methods, spectral multidomain techniques, and front tracking techniques. There is also a paper on adaptive grids. The fluid dynamics papers treat the classical problems of imcompressible flows in helically coiled pipes, vortex breakdown, and transonic flows.

South, J. C., Jr.↗
NASA NTRS1986

Kinetic theory based new upwind methods for inviscid compressible flows

Two new upwind methods called the Kinetic Numerical Method (KNM) and the Kinetic Flux Vector Splitting (KFVS) method for the solution of the Euler equations have been presented. Both of these methods can be regarded as some suitable moments of an upwind scheme for the solution of the Boltzmann equation provided the distribution function is Maxwellian. This moment-method strategy leads to a unification of the Riemann approach and the pseudo-particle approach used earlier in the development of upwind methods for the Euler equations. A very important aspect of the moment-method strategy is that the new upwind methods satisfy the entropy condition because of the Boltzmann H-Theorem and suggest a possible way of extending the Total Variation Diminishing (TVD) principle within the framework of the H-Theorem. The ability of these methods in obtaining accurate wiggle-free solution is demonstrated by applying them to two test problems.

Deshpande, S. M.↗
NASA NTRS1986

A shock capturing technique for hypersonic, chemically relaxing flows

A fully coupled, shock capturing technique is presented for chemically reacting flows at high Mach numbers. The technique makes use of a total variation diminishing (TVD) dissipation operator which results in sharp, crisp shocks. The eigenvalues and eigenvectors of the fully coupled system, which includes species conversion equations in addition to the gas dynamics equations, are analytically derived for a general reacting gas. Species production terms for a model dissociating gas are introduced and are included in the algorithm. The convective terms are solved using a first-order TVD scheme while the source terms are solved using a fourth-order Runge-Kutta scheme to enhance stability. Results from one-dimensional numerical experiments are shown for a two species and a three species gas.

Eberhardt, S.↗
NASA NTRS1986

Comparison of Euler and full potential marching techniques for flows over complex configurations

Two recently developed aerodynamic prediction techniques based on the steady full potential equation and the unsteady Euler equations have been applied to a variety of three-dimensional supersonic flow problems exhibiting embedded subsonic regions. Both techniques utilize planar Gauss-Seidel relaxation in the marching direction and approximate factorization in the cross-flow plane. A conservative switching scheme and flux bias technique are employed in the full potential method to transition from the supersonic marching procedure to a subsonic relaxation algorithm and vice versa. A new unified approach with finite volume, high accuracy (up to third order) Total Variation Diminishing formulation (based on Roe's scheme) is used in the Euler solver. In the supersonic regions of the flow an 'infinitely large' time step is employed, and a finite time step is applied in the subsonic regions of the flow to reach the steady-state as a time-asymptote. Numerical solutions are obtained for a number of complex configurations, including: (1) an elliptic waverider, (2) a realistic fighter configuration, (3) the Space Shuttle, and (4) a Shuttle-like configuration. Both the Full Potential and Euler numerical results are in good agreement with available experimental data.

Szema, K. Y.↗
NASA NTRS1986

On a large time-step high resolution scheme

This paper presents a class of new second-order accurate (2K + 3)-point explicit schemes for the computation of weak solutions of hyperbolic conservation laws, that are total-variation-diminishing under a Courant-Friedrichs-Lewy restriction of K. These highly nonlinear schemes are obtained by applying a nonoscillatory first-order accurate (2K + 1)-point scheme to a modified flux. The derived second-order accurate schemes achieve high resolutions, while retaining the robustness of the original first-order accurate scheme.

Harten, A.↗
NASA NTRS1986

On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations

The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.

Deshpande, S. M.↗
NASA NTRS1987

Development of upwind schemes for the Euler equations

Described are many algorithmic and computational aspects of upwind schemes and their second-order accurate formulations based on Total-Variation-Diminishing (TVD) approaches. An operational unification of the underlying first-order scheme is first presented encompassing Godunov's, Roe's, Osher's, and Split-Flux methods. For higher order versions, the preprocessing and postprocessing approaches to constructing TVD discretizations are considered. TVD formulations can be used to construct relaxation methods for unfactored implicit upwind schemes, which in turn can be exploited to construct space-marching procedures for even the unsteady Euler equations. A major part of the report describes time- and space-marching procedures for solving the Euler equations in 2-D, 3-D, Cartesian, and curvilinear coordinates. Along with many illustrative examples, several results of efficient computations on 3-D supersonic flows with subsonic pockets are presented.

Chakravarthy, Sukumar R.↗
NASA NTRS1986

A second-order accurate kinetic-theory-based method for inviscid compressible flows

An upwind method for the numerical solution of the Euler equations is presented. This method, called the kinetic numerical method (KNM), is based on the fact that the Euler equations are moments of the Boltzmann equation of the kinetic theory of gases when the distribution function is Maxwellian. The KNM consists of two phases, the convection phase and the collision phase. The method is unconditionally stable and explicit. It is highly vectorizable and can be easily made total variation diminishing for the distribution function by a suitable choice of the interpolation strategy. The method is applied to a one-dimensional shock-propagation problem and to a two-dimensional shock-reflection problem.

Deshpande, Suresh M.↗
NASA NTRS1987

A generalized procedure for constructing an upwind based TVD scheme

A generalized formulation for constructing second- and higher-order accurate TVD (total variation diminishing) schemes is presented. A given scheme is made TVD by limiting antidiffusive flux differences with some linear functions, so-called limiters. The general idea of the formulation and its mathematical proof of Harten's TVD conditions is shown by applying the Lax-Wendroff method to scalar nonlinear equations and a constant-coefficient system of conservation laws. For the system of equations, several definitions are derived for the argument used in the limiter function and present their performance in numerical experiments. The formulation is extended to the nonlinear system. It is demonstrated that the present procedure can easily convert existing central or upwind, and second- or higher-order differencing schemes to preserve monotonicity and yield physically admissible solutions. The formulation is simple mathematically as well as numerically; both matrix-vector multiplication and Riemann solver are avoided. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made.

Liou, Meng-Sing↗
NASA NTRS1986

Smoothing and the second law

The technique of obtaining second order, oscillation free, total variation diminishing (TVD), scalar difference schemes by adding a limited diffusion flux (smoothing) to a second order centered scheme is explored. It is shown that such schemes do not always converge to the correct physical answer. The approach presented here is to construct schemes that numerically satisfy the second law of thermodynamics on a cell by cell basis. Such schemes can only converge to the correct physical solution and in some cases can be shown to be TVD. An explicit scheme with this property and second order spatial accuracy was found to have an extremely restrictive time step limitation (Delta t less than Delta x squared). Switching to an implicit scheme removed the time step limitation.

Merriam, Marshal L.↗
NASA NTRS1985

Numerical solution of the unsteady Navier-Stokes equation

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

Osher, Stanley J.↗
NASA NTRS1986

On symmetric and upwind TVD schemes

The performance of the upwind and symmetric total variation diminishing (TVD) schemes in viscous and inviscid airfoil steady-state calculations is considered, and the extension of the implicit second-order-accurate TVD scheme for hyperbolic systems of conservative laws in curvilinear coordinates is discussed. For two-dimensional steady-state applications, schemes are implemented in a conservative noniterative alternating direction implicit form, and results illustrate that the algorithm produces a fairly good solution for an RAE2822 airfoil calculation. The study demonstrates that the symmetric TVD scheme is as accurate as the upwind TVD scheme, while requiring less computational effort than it.

Yee, H. C.↗
NASA NTRS1985

On a class of TVD schemes for gas dynamic calculations

The purpose of this paper is to review a class of explicit and implicit second-order accurate Total Variation Diminishing (TVD) schemes and to show by numerical experiments, the performance of these schemes to the Euler equations of gas dynamics. The method of constructing these second-order accurate TVD schemes is sometimes known as the modified flux approach.

Yee, H. C.↗
NASA NTRS1987

Numerical simulation of shock wave diffraction by TVD schemes

An upwind total variation diminishing (TVD) scheme and a predictor-corrector symmetric TVD scheme were used to numerically simulate the blast wave diffraction on a stationary object. The objective is to help design an optimum configuration so that lateral motion is minimized and at the same time vortex shedding and flow separation are reduced during a blast wave encounter. Results are presented for a generic configuration for both a coarse grid and a fine grid to illustrate the global and local diffraction flow fields. Numerical experiments for the shock wave reflection on a wedge are also included to validate the current approach. Numerical study indicated that these TVD schemes are more stable and produced higher shock resolution than classical shock capturing methods such as the explicit MacCormack scheme.

Young, Victor Y. C.↗