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Gottlieb, D.

Publications and source records attributed to Gottlieb, D..

34 records · Page 2
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NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗
NASA NTRS1983

Improving the convergence rate of parabolic ADI methods

The rate of convergence to steady state of parabolic Alternating Direction Implicit (ADI) solvers is analyzed in terms of the L(2)-norms of the residuals. The analysis allows one to predict the number of iterations necessary for convergence as function of the Courant number, Lambda. A simple modification of existing ADI codes is devised. It improves the convergence rate substantially and is insensitive to the Courant number in a large range of Lambda.

Abarbanel, S. S.↗
NASA NTRS1981

Stability of pseudospectral and finite-difference methods for variable coefficient problems

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.

Gottlieb, D.↗
NASA NTRS1981

Optimal time splitting for two- and three-dimensional Navier-Stokes equations with mixed derivatives

A new explicit, time splitting algorithm has been developed for finite difference modelling of the full two and three-dimensional time-dependent, compressible, viscous Navier-Stokes equations of fluid mechanics. The scheme is optimal in the sense that the split operators achieve their maximum allowable time step, i.e., the corresponding Courant number. The algorithm allows a conservation-form formulation. Stability is proven analytically and verified numerically. In proving stability it was shown that all nine matrix coefficients of the Navier-Stokes equations are simultaneously symmetrizable by a similarity transformation. Two such transformations and their resulting symmetric matrix coefficients are presented explicitly.

Abarbanel, S.↗
NASA NTRS1981

The stability of pseudospectral-Chebyshev methods

The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.

Gottlieb, D.↗
NASA NTRS1980

On time discretizations for spectral methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.

Gottlieb, D.↗
NASA NTRS1978

Boundary conditions for multistep finite-difference methods for time-dependent equations

The stability and accuracy of various boundary treatments are analyzed for the two-step Richtmyer and MacCormack methods. Special attention is paid to ways of imposing the extra boundary conditions after the first step of the two-step process. The theory of Kreiss is used to study stability properties for both scalar and vector equations. The theory of Skollermo is used to compare accuracies of the various methods. Computations were also performed on both wavelike equations and on systems that approach a steady state. Several suggestions are given for more reliable boundary treatments.

Gottlieb, D.↗
NASA NTRS1978

On acceleration of MacCormack's scheme

An acceleration of MacCormack's scheme due to Desideri and Tannehill is analyzed. It is found that for hyperbolic problems one cannot improve upon the efficiency of MacCormack's method. For parabolic problems the time step can be chosen arbitrarily large without loss of stability by an appropriate choice of the acceleration parameters. When applied to the heat equation this method is equivalent to both the Dufort-Frankel scheme and to MSOR.

Gottlieb, D.↗
NASA NTRS1976

A note on the leap-frog scheme in two and three dimensions

The paper considers the leap-frog finite-difference method (Kreiss and Oliger, 1973) for systems of partial differential equations of the form du/dt = dF/dx + dG/dy + dH/dz, where d denotes partial derivative, u is a q-component vector and a function of x, y, z, and t, and the vectors F, G, and H are functions of u only. The original leap-frog algorithm is shown to admit a modification that improves on the stability conditions for two and three dimensions by factors of 2 and 2.8, respectively, thereby permitting larger time steps. The scheme for three dimensions is considered optimal in the sense that it combines simple averaging and large time steps.

Abarbanel, S.↗
NASA NTRS1976

On the Navier-Stokes equations with constant total temperature

For various applications in fluid dynamics, one can assume that the total temperature is constant. Therefore, the energy equations can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed in this paper. It is shown that the system is strictly hyperbolic and well posed for the initial-value problem. Boundary conditions are described such that the linearized system is well posed. The hopscotch method is investigated and numerical results are presented.

Gottlieb, D.↗
NASA NTRS1976

Multidimensional difference schemes with fourth-order accuracy

An explicit finite-difference algorithm is presented for the solution of quasilinear divergence free multidimensional hyperbolic systems. The method consists of four steps per time level. The resulting scheme is fourth-order accurate in both space and time, though the intermediate steps are only first-order accurate. The family of schemes introduced is dissipative, and hence, suitable for both smooth flows and flows containing shocks. This method is compared, in several numerical examples, with both second-order schemes and others that are fourth order in space, but second order in time.

Turkel, E.↗
NASA NTRS1976

Generalized Du Fort-Frankel methods for parabolic initial-boundary value problems

The Du Fort-Frankel difference scheme is generalized to difference operators of arbitrary high order accuracy in space and to arbitrary order of the parabolic differential operator. Spectral methods can also be used to approximate the spatial part of the differential operator. The scheme is explicit, and it is unconditionally stable for the initial value problem. Stable boundary conditions are given for two different fourth order accurate space approximations.

Gottlieb, D.↗
NASA NTRS1975

Generalized Du Fort-Frankel methods for parabolic initial boundary value problems

The Du Fort-Frankel difference scheme is generalized to difference operators of arbitrary high order accuracy in space and to arbitrary order of the parabolic differential operator. Spectral methods can also be used to approximate the spatial part of the differential operator. The scheme is explicit, and it is unconditionally stable for the initial value problem. Stable boundary conditions are given for two different fourth order accurate space approximations.

Gottlieb, D.↗
NASA NTRS1975

On the Navier-Stokes equations with constant total temperature

For various applications in fluid dynamics, it is assumed that the total temperature is constant. Therefore, the energy equation can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed. It is shown that the system is strictly hyperbolic and well posed for the initial value problems. Boundary conditions are described such that the linearized system is well posed. The Hopscotch method is investigated and numerical results are presented.

Gottlieb, D.↗