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NASA NTRS1992

Accurate upwind-monotone (nonoscillatory) methods for conservation laws

The well known MUSCL scheme of Van Leer is constructed using a piecewise linear approximation. The MUSCL scheme is second order accurate at the smooth part of the solution except at extrema where the accuracy degenerates to first order due to the monotonicity constraint. To construct accurate schemes which are free from oscillations, the author introduces the concept of upwind monotonicity. Several classes of schemes, which are upwind monotone and of uniform second or third order accuracy are then presented. Results for advection with constant speed are shown. It is also shown that the new scheme compares favorably with state of the art methods.

Huynh, Hung T.↗
NASA NTRS1992

A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting

The present treatment of elliptic regions via hyperbolic flux-splitting and high order methods proposes a flux splitting in which the corresponding Jacobians have real and positive/negative eigenvalues. While resembling the flux splitting used in hyperbolic systems, the present generalization of such splitting to elliptic regions allows the handling of mixed-type systems in a unified and heuristically stable fashion. The van der Waals fluid-dynamics equation is used. Convergence with good resolution to weak solutions for various Riemann problems are observed.

Shu, Chi-Wang↗
NASA NTRS1995

Adaptively-refined overlapping grids for the numerical solution of systems of hyperbolic conservation laws

Adaptive mesh refinement (AMR) in conjunction with higher-order upwind finite-difference methods have been used effectively on a variety of problems in two and three dimensions. In this paper we introduce an approach for resolving problems that involve complex geometries in which resolution of boundary geometry is important. The complex geometry is represented by using the method of overlapping grids, while local resolution is obtained by refining each component grid with the AMR algorithm, appropriately generalized for this situation. The CMPGRD algorithm introduced by Chesshire and Henshaw is used to automatically generate the overlapping grid structure for the underlying mesh.

Brislawn, Kristi D.↗
NASA NTRS1998

The Space-Time Conservation Element and Solution Element Method-A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws: Numerical Simulation of Shock Waves and Contact Discontinuities - 2

Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes described previously (in Part I) are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing scheme, the CPU time is about 2 micro-secs per mesh point per marching step on a Cray C90 machine), the numerical results, when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these schemes can accurately resolve shock and contact discontinuities consistently.

Wang, Xiao-Yen↗
NASA NTRS1995

Numerical Solution of Multi-Dimensional Hyperbolic Conservation Laws on Unstructured Meshes

The lecture material will discuss the application of one-dimensional approximate Riemann solutions and high order accurate data reconstruction as building blocks for solving multi-dimensional hyperbolic equations. This building block procedure is well-documented in the nationally available literature. The relevant stability and convergence theory using positive operator analysis will also be presented. All participants in the minisymposium will be asked to solve one or more generic test problems so that a critical comparison of accuracy can be made among differing approaches.

Barth, Timothy J.↗
NASA NTRS2003

On a Non-Reflecting Boundary Condition for Hyperbolic Conservation Laws

A non-reflecting boundary condition (NRBC) for practical computations in fluid dynamics and aeroacoustics is presented. The technique is based on the first principle of non-reflecting, plane wave propagation and the hyperbolicity of the Euler equation system. The NRBC is simple and effective, provided the numerical scheme maintains locally a C(sup 1) continuous solution at the boundary. Several numerical examples in 1D, 2D, and 3D space are illustrated to demonstrate its robustness in practical computations.

Loh, Ching Y.↗
NASA NTRS2014

On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

Huynh, H. T.↗
NASA NTRS2003

On a Non-Reflecting Boundary Condition for Hyperbolic Conservation Laws

A non-reflecting boundary condition (NRBC) for practical computations in fluid dynamics and aeroacoustics is presented. The technique is based on the hyperbolicity of the Euler equation system and the first principle of plane (simple) wave propagation. The NRBC is simple and effective, provided the numerical scheme maintains locally a C(sup 1) continuous solution at the boundary. Several numerical examples in ID, 2D and 3D space are illustrated to demonstrate its robustness in practical computations.

Loh, Ching Y.↗
NASA NTRS1984

Kinematics of the CS method for the treatment of molecular collisions

The problem of the quantum treatment of nonreactive collisions of two simple molecules or of an atom and a molecule is considered mathematically. Kinematical structure theorems analogous to the Wigner-Eckart theorem for scalar operators are derived for the transition operator (T-operator) of a two-molecule system, where, in addition to the usual conservation laws, certain additional conservation laws are presumed satisfied in a collision. The additional conservation laws were examined in order to develop a testable approximation scheme (CS method) for the specification of the kinematics of a system of two rigid diatomic molecules. The rand p-helicity basis states for a two-rigid rotor system are defined and unitary transformations established between these and conventional bases. It is found that only questionable criteria exist for testing the on-the-energy-shell matrix of a T(energy) operator for the presence of properties of two of the additional conservation laws.

Hahne, G. E.↗
NASA NTRS1991

Choice of velocity variables for complex flow computation

The issue of adopting the velocity components as dependent velocity variables for the Navier-Stokes flow computations is investigated. The viewpoint advocated is that a numerical algorithm should preferably honor both the physical conservation law in differential form and the geometric conservation law in discrete form. With the use of Cartesian velocity vector, the momentum equations in curvilinear coordinates can retain the full conservation-law form and satisfy the physical conservation laws. With the curvilinear velocity components, source terms appear in differential equations and hence the full conservation law form can not be retained. In discrete expressions, algorithms based on the Cartesian components can satisfy the geometric conservation-law form for convection terms but not for viscous terms; those based on the curvilinear components, on the other hand, cannot satisfy the geometric conservation-law form for either convection or viscous terms. Several flow solutions for domain with 90 and 360 degree turnings are presented to illustrate the issues of using the Cartesian velocity components and the staggered grid arrangement.

Shyy, W.↗
NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Chapter 1 briefly reviews several related topics associated with the symmetrization of systems of conservation laws and quasi-conservation laws: (1) Basic Entropy Symmetrization Theory; (2) Symmetrization and eigenvector scaling; (3) Symmetrization of the compressible Navier-Stokes equations; and (4) Symmetrization of the quasi-conservative form of the magnetohydrodynamic (MHD) equations. Chapter 2 describes one of the best known tools employed in the study of differential equations, the maximum principle: any function f(x) which satisfies the inequality f(double prime)>0 on the interval [a,b] attains its maximum value at one of the endpoints on the interval. Chapter three examines the upwind finite volume schemes for scalar and system conservation laws. The basic tasks in the upwind finite volume approach have already been presented: reconstruction, flux evaluation, and evolution. By far, the most difficult task in this process is the reconstruction step.

Bart, Timothy J.↗
NASA NTRS1990

Shock-layer bounds for a singularly perturbed equation

The size of the shock-layer governed by a conservation law is studied. The conservation law is a parabolic reaction-convection-diffusion equation with a small parameter multiplying the diffusion term and convex flux. Rigorous upper and lower bounding functions for the solution of the conservation law are established based on maximum-principle arguments. The bounding functions demonstrate that the size of the shock-layer is proportional to the parameter multiplying the diffusion term.

Scroggs, Jeffrey S.↗
NASA NTRS1946

Some Effects of Compressibility on the Flow Through Fans and Turbines

The laws of conservation of mass, momentum, and energy are applied to the compressible flow through a two-dimensional cascade of airfoils. A fundamental relation between the ultimate upstream and downstream flow angles, the inlet Mach number, and the pressure ratio across the cascade is derived. Comparison with the corresponding relation for incompressible flow shows large differences. The fundamental relation reveals two ranges of flow angles and inlet Mach numbers, for which no ideal pressure ratio exists. One of these nonideal operating ranges is analogous to a similar type in incompressible flow. The other is characteristic only of compressible flow. The effect of variable axial-flow area is treated. Some implications of the basic conservation laws in the case of nonideal flow through cascades are discussed.

Perl, W.↗
NASA NTRS1987

Lectures series in computational fluid dynamics

The lecture notes cover the basic principles of computational fluid dynamics (CFD). They are oriented more toward practical applications than theory, and are intended to serve as a unified source for basic material in the CFD field as well as an introduction to more specialized topics in artificial viscosity and boundary conditions. Each chapter in the test is associated with a videotaped lecture. The basic properties of conservation laws, wave equations, and shock waves are described. The duality of the conservation law and wave representations is investigated, and shock waves are examined in some detail. Finite difference techniques are introduced for the solution of wave equations and conservation laws. Stability analysis for finite difference approximations are presented. A consistent description of artificial viscosity methods are provided. Finally, the problem of nonreflecting boundary conditions are treated.

Thompson, Kevin W.↗
NASA NTRS1978

The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology

A program is described which is intended to derive a generalized system of gravitational equations that allow (but do not require) G to vary, to use the 3-K blackbody radiation to fix the relation between G and the gauge function, and to employ Dirac's (1937) Large Numbers Hypothesis to derive the geometry of the universe. Einstein's equations are retained in their total integrity, but the specification is made that they are valid only when gravitational units are used. A scale-invariant form of Einstein's equations is obtained, and from this are derived the energy conservation law, the baryon-number conservation law, and the appropriate cosmological equations. Dirac's proposals of 1937 and 1973 are incorporated into the formalism, and a gauge based on consolidation of the 3-K blackbody radiation is presented. A unique solution for the geometry of the universe is determined for zero curvature solely from the 3-K radiation and the Large Numbers Hypothesis; this solution predicts a deceleration parameter exactly equal to unity.

Canuto, V.↗
NASA NTRS1999

Acoustic Energy Estimates in Inhomogeneous Moving Media

In ducted fan engine noise research, there is a need for defining a simple and easy to use acoustic energy conservation law to help in quantification of noise control techniques. There is a well known conservation law relating acoustic energy and acoustic energy flux in the case of an isentropic irrotational flow. Several different approaches have been taken to generalize this conservation law. For example, Morfey finds an identity by separating out the irrotational part of the perturbed flow. Myers is able to find a series of indentities by observing an algebraic relationship between the basic conservation of energy equation for a background flow and the underlying equations of motion. In an approximate sense, this algebraic relationship is preserved under perturbation. A third approach which seems to have not been pursued in the literature is a result known as Noether's theorem. There is a Lagrangian formulation for the Euler equation of fluid mechanics. Noether's theorem says that any group action that leaves the Lagrangian action invariant leads to a conserved quantity. This presentation will include a survey of current results regarding acoustic energy and preliminary results on the symmetries of the Lagrangian.

Farassat, F.↗
NASA NTRS1983

A numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution

A numerical method is presented for studying one-dimensional electron plasma evolution under typical interplanetary conditions. The method applies the Fourier-Fourier transform approach to a plasma model that is a generalization of the electrostatic Vlasov-Poisson system of equations. Conservation laws that are modified to include the plasma model generalization and also the boundary effects of nonperiodic solutions are given. A new conservation law for entropy in the transformed space is then introduced. These conservation laws are used to verify the numerical solutions. A discretization error analysis is presented. Two numerical instabilities and the methods used for their suppression are treated. It is shown that in interplanetary plasma conditions, the bump-on-tail instability produces significant excitation of plasma oscillations at the Bohm-Gross frequency and its second harmonic. An explanation of the second harmonic excitation is given in terms of wave-wave coupling during the growth phase of the instability.

Klimas, A. J.↗