Robert F. Warming

Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods☆

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

An implicit finite difference algorithm for hyperbolic systems in conservation law form

Abstract An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.

Radiative transport and wall temperature slip in an absorbing planar medium

Abstract Radiative heat transfer through a nonisothermal absorbing and emitting grey gas between heated walls is studied. Specific attention is directed toward the evaluation of temperature near the walls and of the precise evaluation of energy flux by means of methods and tabulated functions studied by Chandrasekhar and Ambartsumian. The precision achieved permits an assessment of the accuracy of existing approximate methods and of the errors incurred in numerical solutions of the governing equations.

Diagonalization and simultaneous symmetrization of the gas-dynamic matrices

The hyperbolic nature of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation for diagonalizing an arbitrary linear combination of coefficient matrices. It is shown that the individual matrices are simultaneously symmetrized by the similarity transformation. The transformations and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.

Alternating Direction Implicit Methods for Parabolic Equations with a Mixed Derivative

Alternating direction implicit (ADI) schemes for two-dimensional parabolic equations with a mixed derivative are constructed by using the class of all

Theoretical predictions of radiative transfer in a homogenous cylindrical medium

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An extension ofA-stability to alternating direction implicit methods

-stable linear two-step methods in conjunction with the method of approximate factorization. The mixed derivative is treated with an explicit two-step method which is compatible with an implicit

Application of invariance principles to a radiative transfer problem in a homogeneous spherical medium

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The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

-stable method. The parameter space for which the resulting ADI schemes are second-order accurate and unconditionally stable is determined. Some numerical examples are given.

Theoretical predictions of spectral line formation by noncoherent scattering.

Abstract Analytical and numerical methods of predicting radiative transfer are developed particularly for a homogeneous medium with constant extinction coefficient in a circularly symmetric, cylindrical region. Efficient formulations of the influence functions in the governing integral equation and the flux integral are given. Applications of the results include an integral method of calculating flux losses associated with radial distributions of internal sources of energy release, and predictions based on the use of simplified methods of approximation. Planar and spherical regions are also considered since comparisons between approximate and precise predictions then become possible. Cases of physical interest include both conservative and noncervative radiative transfer; for example, thermal radiation in the continuum regime as well as spectral line emission from an isothermal plasma with two-level impurity ions.