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NASA NTRS1982

Multiple-grid acceleration of Lax-Wendroff algorithms

A technique for accelerating the convergence of a one-step Lax-Wendroff method to a steady-state solution is discussed and its applicability extended to the more general class of two-step Lax-Wendroff methods. Several two-step methods which lead to quite efficient multiple grid algorithms are discussed. Computational results are presented using the full two dimensional Euler equations for both subcritical and shocked supercritical flows. Extensions and generalizations are mentioned.

Johnson, G. M.↗

NASA NTRS1984

A numerical transport scheme which avoids negative mixing ratios

A new scheme for numerically integrating the transport equation has been developed. This square root method avoids the problem of negative mixing ratios by using the square root of the concentration instead of the concentration itself as an advective time step. Conservation of total mass at every time step is achieved by extending the concept of quadratic conservation to the numerical time integration method. A few time schemes that fulfill the requirement of step-by-step quadratic conservation are described, the simplest of which is the modified Lax-Wendroff method.

Schneider, H.-R.↗

NASA NTRS1974

A numerical calculation of outward propagation of solar disturbances

The responses of the solar atmosphere due to an outward propagation shock are examined by employing the Lax-Wendroff method to solve the set of nonlinear partial differential equations in the model of the solar atmosphere. It is found that this theoretical model can be used to explain the solar phenomena of surge and spray. A criterion to discriminate the surge and spray is established and detailed information concerning the density, velocity, and temperature distribution with respect to the height and time is presented. The complete computer program is also included.

Wu, S. T.↗

NASA NTRS1974

Propagation of solar disturbances in interplanetary space

Time-dependent solutions of a one-fluid model of the interplanetary medium are investigated. This set of unsteady hydrodynamic equations has been written in conservation form in order to apply the Lax-Wendroff method for the solution of this problem. The initial condition is specified by a pulse at 1 solar radius. The equilibrium condition is chosen to be the steady solution of a quiet solar wind. The specified solar disturbances in this calculation are allowed to be both sub- and supersonic by the present theoretical formulation. The results are presented in terms of density, velocity, and temperature profiles of the interplanetary gas flow at heliocentric distances up to about 10 AU at any particular time. The trajectories of disturbances for various initial pulses are shown. Some 1972 solar-flare observational data are compared with these theoretical calculations. From these calculations, the effects on the interplanetary environment, due to the propagation of solar disturbances, can be determined.

Wu, S. T.↗

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 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 nonlinear 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 a scalar nonlinear equation and 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 to numerical experiments. Then the formulation is formally extended to the nonlinear system of equations. It is demonstrated that use of the present procedure allows easy conversion of existing central or upwind, and second- or higher-order differencing schemes so as to preserve monotonicity and to yield physically admissible solutions. The formulation is simple mathematically as well as numerically; neither matrix-vector multiplication nor Riemann solver is required. Roughly twice as much computational effort is needed as compared to conventional scheme. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made. Numerical examples including various ranges of problems show both time- and spatial-accuracy in comparison with exact solutions.

Liou, Meng-Sing↗

NASA NTRS1988

Magnetic fields interacting with nonlinear compressible convection

Two-dimensional numerical simulations are used to study fully compressible convection in the presence of an imposed magnetic field. Highly nonlinear flows are considered that span multiple density scale heights. The convection tends to sweep the initially uniform vertical magnetic field into concentrated flux sheets with significant magnetic pressures. These flux sheets are partially evacuated, and effects of buoyancy and Lorentz forces there can serve to suppress motions. The flux sheets can be surrounded by a sheath of descending flow. If the imposed magnetic field is sufficiently strong, the convection can become oscillatory. The unstably stratified fluid layer has an initial density ratio (bottom to top of layer) of 11. Surveys of solutions at fixed Rayleigh number sample Chandrasekhar numbers from 1 to 1000 and magnetic Prandtl numbers from 1/16 to 1. These nonlinear simulations utilize a two-dimensional numerical scheme based on a modified two-step Lax-Wendroff method.

Hulburt, Neal E.↗

NASA NTRS1986

Report on the feasibility of hypercube concurrent processing systems in computational fluid dynamics

The feasibility of using hypercube-connected concurrent processor systems for problems in computational fluid dynamics is studied. Both explicit and implicit numerical methods are considered and several alternative implementations of these methods are evaluated on concurrent processor systems. A Lax-Wendroff explicit method was designed and implemented for the Navier-Stokes equations. The code runs on the Intel iPSC concurrent processor system. Tests of this code show that it is reasonably efficient. The Beam and Warming implicit factored method was designed and implemented for Berger's equation. Preliminary tests show that the efficiency of code is poor.

Bruno, J.↗

NASA NTRS1992

Unstructured grid methods for compressible flows

The implementation of the finite element method on unstructured triangular grids is described and the development of centered finite element schemes for the solution of the compressible Euler equation on general triangular and tetrahedral grids is discussed. Explicit and implicit Lax-Wendroff type methods and a method based upon the use of explicit multistep timestepping are considered. In the latter case, the convergence behavior of the method is accelerated by the incorporation of a fully unstructured multigrid procedure. The advancing front method for generating unstructured grids of triangles and tetrahedra is described and the application of adaptive mesh techniques to both steady and transient flow analysis is illustrated.

Morgan, K.↗

NASA NTRS1976

Interplanetary disturbances in the solar wind produced by density, temperature, or velocity pulses at 0.08 AU

Time-dependent solutions of a one-fluid model of the interplanetary medium are investigated. This set of unsteady hydrodynamic equations has been written in conserved form in order to apply the Lax-Wendroff (1960) method for the solution of this problem. The initial disturbance is specified by a pulse at 0.08 AU. Physically, this pulse can be interpreted as having been caused by a solar flare, surge, or any other solar disturbance. The equilibrium condition is determined to be the steady solution of the governing equations and represents the quiet solar wind. Results are presented in terms of density, temperature, and velocity profiles of the interplanetary gas flow at heliocentric distances up to 6 AU at several times. Also, the trajectories of disturbances for various initial pulses are shown. Some June 1972 interplanetary observational data are compared with these theoretical calculations. On the basis of these results, the effects of solar disturbances on the interplanetary environment (such as the generation of large nonlinear wave trains in the shock wakes) can be inferred.

Wu, S. T.↗

NASA NTRS1979

Numerical study of an explosion in a non-homogeneous medium with and without magnetic fields

A version of the two step Lax-Wendroff difference method with second order accuracy is used to seek solutions of the unsteady magnetohydrodynamic (MHD) equations for the study of an explosion in a non-homogeneous medium with and without magnetic fields. The explosion is initiated by introducing a finite amount of energy in a small volume of gas which leads to an instantaneous increase of gas temperature. Two types of magnetic field configurations (i.e. open and closed) are considered to illustrate the dependence and differences of magnetic and a non-magnetic flow motion. Numerical results show the strong dependence of induced MHD flow field on the magnetic field configuration and strength. The development and decay of the fast and slow MHD shock waves, as well as the ordinary gasdynamic shock waves, are also represented well by the present computer simulations. In conclusion, we have demonstrated that the numerical scheme we have used is a reliable one for studying 2-dimensional MHD problems in non-homogeneous medium.

Han, S. M.↗

NASA NTRS1983

Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm

A multiple-grid algorithm for use in efficiently obtaining steady solutions to the Euler and Navier-Stokes equations is presented. The convergence of the explicit MacCormack algorithm on a fine grid is accelerated by propagating transients from the domain using a sequence of successively coarser grids. Both the fine and coarse grid schemes are readily vectorizable. The combination of multiple-gridding and vectorization results in substantially reduced computational times for the numerical solution of a wide range of flow problems. Results are presented for subsonic, transonic, and supersonic inviscid flows and for subsonic attached and separated laminar viscous flows. Work reduction factors over a scalar, single-grid algorithm range as high as 76.8.

Navier Stokes simulation↗

NASA NTRS1987

Construction of explicit and implicit symmetric TVD schemes and their applications

A one-parameter family of explicit and implicit total variation diminishing (TVD) schemes is developed which permits incorporation of an expanded group of slope and flux limiters. The numerical technique is intended for use in calculations which include a time-differencing scheme and an optional Lax-Wendroff scheme. Methods of extending the TVD models to nonlinear scalar equations and systems of hyperbolic conservation equations are described. Sample results are presented from calculations of shocked flows around NACA 0012 and NACA 0018 airfoils.

Yee, H. C.↗

NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗

NASA NTRS1988

A Three-Dimensional, Time-Dependent Numerical Modeling of Super-Sonic, Super-Alfvenic MHD Flow

A three-dimensional, time-dependent, MHD disturbance propagating through a steady inhomogeneous MHD flow is analyzed by a numerical model based on the Lax-Wendroff finite difference method. The initial steady MHD flow is assumed to be supersonic and super-Alfvenic. The background steady MHD flow is obtained by dropping the time-dependent terms in the unsteady MHD equations. A transient MHD disturbance corresponding to a MHD shock is generated at the lower radial boundary surface. The resulting development and propagation of 3-D MHD disturbances interacting with the background steady MHD flow are numerically simulated in this first application to the field of interplanetary physics. Detailed descriptions of the mathematical formulation and numerical methodology are presented. Example computations are performed to demonstrate the validity of the formulation and the capabilities of the model.

Han, S. M.↗

NASA NTRS1985

Stability of interfaces with mesh refinement

A study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, a root condition which implies stability for the initial-boundary value problem for this type of interface is derived. From the root condition, the stability of several interface equations is proved, using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, stability for Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved.

Berger, M. J.↗

NASA NTRS1992

Finite-difference algorithms for the time-domain Maxwell's equations - A numerical approach to RCS analysis

The applications of two CFD-based finite-difference methods to computational electromagnetics are investigated. In the first method, the time-domain Maxwell's equations are solved using the explicit Lax-Wendroff scheme and in the second method, the second-order wave equations satisfying the Maxwell's equations are solved using the implicit Crank-Nicolson scheme. The governing equations are transformed to a generalized curvilinear coordinate system and solved on a body-conforming mesh using the scattered-field formulation. The induced surface current and the bistatic radar cross section are computed and the results are validated for several two-dimensional test cases involving perfectly-conducting scatterers submerged in transverse-magnetic plane waves.

Vinh, Hoang↗

NASA NTRS1991

Calculation of unsteady linearized Euler flows in cascades using harmonically deforming grids

A method for calculating unsteady, inviscid, compressible flows in cascades is presented. Using the linearized Euler technique, the flow is decomposed into a steady or mean flow plus a harmonically varying small disturbance flow. The equations that describe the small disturbance flow are linear variable coefficient equations, and are solved using a pseudo-time time marching Lax-Wendroff technique. Unlike previous linearized methods, however, the solution is computed on a harmonically deforming computational grid that conforms to the motion of the vibrating airfoils. The mean flow and perturbation flow solutions are defined in the deforming coordinate system rather than in a coordinate system fixed in space. Hence, no extrapolation terms are required to implement the upwash boundary conditions at the airfoil surfaces, significantly improving the accuracy of the method. For transonic flow calculations, unsteady shock motions are modelled using shock capturing. The unsteady loads due to the shock motion are then seen as pressure impulses. Representative computational results are presented for transonic channel flows and subsonic and transonic cascade flows.

Hall, Kenneth C.↗