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Report presents comparative study of several high-resolution explicit numerical-simulation schemes capturing shocks in one-dimensional flows of real gas. One-dimensional schemes compared with respect to: ability to capture shocks, resolution of shocks, overall accuracy, and computational efficiency.
The goal is to extend our adaptive numerical dissipation control in high order filter schemes and our new divergence-free methods for ideal MHD to non-ideal MHD that include viscosity and resistivity. The key idea consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free of numerical dissipation contamination. These scheme-independent detectors are capable of distinguishing shocks/shears, flame sheets, turbulent fluctuations and spurious high-frequency oscillations. The detection algorithm is based on an artificial compression method (ACM) (for shocks/shears), and redundant multi-resolution wavelets (WAV) (for the above types of flow feature). These filter approaches also provide a natural and efficient way for the minimization of Div(B) numerical error. The filter scheme consists of spatially sixth order or higher non-dissipative spatial difference operators as the base scheme for the inviscid flux derivatives. If necessary, a small amount of high order linear dissipation is used to remove spurious high frequency oscillations. For example, an eighth-order centered linear dissipation (AD8) might be included in conjunction with a spatially sixth-order base scheme. The inviscid difference operator is applied twice for the viscous flux derivatives. After the completion of a full time step of the base scheme step, the solution is adaptively filtered by the product of a 'flow detector' and the 'nonlinear dissipative portion' of a high-resolution shock-capturing scheme. In addition, the scheme independent wavelet flow detector can be used in conjunction with spatially compact, spectral or spectral element type of base schemes. The ACM and wavelet filter schemes using the dissipative portion of a second-order shock-capturing scheme with sixth-order spatial central base scheme for both the inviscid and viscous MHD flux derivatives and a fourth-order Runge-Kutta method are denoted.
We have developed a new three-dimensional general relativistic magnetohydrodynamic (GRMHD) code, RAISHIN, using a conservative, high resolution shock-capturing scheme. The numerical fluxes are calculated using the Harten, Lax, & van Leer (HLL) approximate Riemann solver scheme. The flux-interpolated, constrained transport scheme is used to maintain a divergence-free magnetic field. In order to examine the numerical accuracy and the numerical efficiency, the code uses four different reconstruction methods: piecewise linear methods with Minmod and MC slope-limiter function, convex essentially non-oscillatory (CENO) method, and piecewise parabolic method (PPM) using multistep TVD Runge-Kutta time advance methods with second and third-order time accuracy. We describe code performance on an extensive set of test problems in both special and general relativity. Our new GRMHD code has proven to be accurate in second order and has successfully passed with all tests performed, including highly relativistic and magnetized cases in both special and general relativity.
The central difference scheme (CDS) is a second order accurate scheme which is free of numerical diffusion (in the second order sense) and is simple to implement: However, for grid Peclet numbers larger than 2, the CDS leads to over- and undershoots and is unstable. The present paper describes a method, called CONDIF, which retains the essential nature of the CDS but eliminates the over- and under-shoots. It leads to unconditionally positive coefficients and, in the limit, approaches the CDS for all values of grid Peclet numbers. The CONDIF modifies the CDS by introducing a controlled amount of numerical diffusion based on the local gradients. In the worst case the scheme yields results similar to those of the hybrid scheme. This paper reports the results obtained from CONDIF for a number of test problems which have been widely used for comparative study of numerical schemes in the published literature. For most of these problems, the CONDIF results are significantly more accurate than the hybrid scheme at high Peclet numbers. In particular, the CONDIF scheme depicts much lower level of numerical diffusion than the hybrid scheme even when the Peclet number is very high and the flow is at large angles to the grid.
The required type and amount of numerical dissipation/filter to accurately resolve all relevant multiscales of complex MHD unsteady high-speed shock/shear/turbulence/combustion problems are not only physical problem dependent, but also vary from one flow region to another. In addition, proper and efficient control of the divergence of the magnetic field (Div(B)) numerical error for high order shock-capturing methods poses extra requirements for the considered type of CPU intensive computations. The goal is to extend our adaptive numerical dissipation control in high order filter schemes and our new divergence-free methods for ideal MHD to non-ideal MHD that include viscosity and resistivity. The key idea consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. These scheme-independent detectors are capable of distinguishing shocks/shears, flame sheets, turbulent fluctuations and spurious high-frequency oscillations. The detection algorithm is based on an artificial compression method (ACM) (for shocks/shears), and redundant multiresolution wavelets (WAV) (for the above types of flow feature). These filters also provide a natural and efficient way for the minimization of Div(B) numerical error.
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.
The application of a flux-spline scheme to three-dimensional fluid flow is reported. A comparison is made of the performance of this scheme with that of the Power-law differencing scheme. The numerical results are compared with experimental data. For the problem considered in this study, the flux-spline scheme is more accurate than the Power-law. For a prescribed accuracy, the flux-spline scheme requires a far fewer number of grid points. Thus, it has the potential of providing a numerical error-free solution, especially for three-dimensional flows, without requiring an excessively fine grid.
This paper extends the finite element scheme to handle the highly nonlinear interfacial fields generated in the fluid filled annulli of squeeze film and journal bearings so as to model the transient response of rotor-bearing-stator systems. Since such simulations are highly nonlinear, direct numerical integration schemes are employed to generate the overall response. In this context, the paper gives consideration to such items as (1) numerical efficiency/stability, (2) comparison of implicit and explicit schemes, (3) determines extent of response nonlinearity as well as (4) extensively benchmarks the overall concept/methodologies.
The essentially non-oscillatory (ENO) finite difference scheme is applied to systems of conservation laws of mixed hyperbolic-elliptic type. A flux splitting, with the corresponding Jacobi matrices having real and positive/negative eigenvalues, is used. The hyperbolic ENO operator is applied separately. The scheme is numerically tested on the van der Waals equation in fluid dynamics. Convergence was observed with good resolution to weak solutions for various Riemann problems, which are then numerically checked to be admissible as the viscosity-capillarity limits. The interesting phenomena of the shrinking of elliptic regions if they are present in the initial conditions were also observed.
The technique of obtaining second-order oscillation-free total -variation-diminishing (TVD), scalar difference schemes by adding a limited diffusive 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 extremely restrictive time-step limitation. Switching to an implicit scheme removed the time-step limitation.
In this work, we present a super-time-stepping scheme for numerically solving parabolic partial differential equations with Dirichlet boundary conditions (BC). Using the general Forward Euler scheme, one can show that by taking varying step sizes there is the potential of propagating the solution forward in time by a greater amount than with uniform step sizes, while maintaining the same order of accuracy. As shown in previous works, if one further requires that the scheme have the Convex Monotone Property (CMP), then there exists a scheme which results in linear, monotone stability of the solution. This monotone stability is highly desirable in many physical situations, such as thermal diffusion, where the physical system will not oscillate, but will behave monotonically. However, the schemes devised in previous works do not include situations that have a boundary condition, and the inclusion of boundary conditions will henceforth be our focus. It is shown that a particular Runge-Kutta-Gegenbauer class of schemes [5] will maintain the CMP even in the presence of Dirichlet BC.
A single nonprecipitating cumulus congestus setup is applied to compare droplet spectra grown by the diffusion of water vapor in Eulerian bin and particle-based Lagrangian microphysics schemes. Bin microphysics represent droplet spectral evolution applying the spectral density function. In the Lagrangian microphysics, computational particles referred to as superdroplets are followed in time and space with each superdroplet representing a multiplicity of natural cloud droplets. The same cloud condensation nuclei (CCN) activation and identical representation of the droplet diffusional growth allow the comparison. The piggybacking method is used with the two schemes operating in a single simulation, one scheme driving the dynamics and the other one piggybacking the simulated flow. Piggybacking allows point-by-point comparison of droplet spectra predicted by the two schemes. Overall, the results show the impact of inherent limitations of the two microphysics simulation methods, numerical diffusion in the Eulerian scheme and a limited number of superdroplets in the Lagrangian scheme. Numerical diffusion in the Eulerian scheme results in a more dilution of the cloud upper half and thus smaller cloud droplet mean radius. The Lagrangian scheme typically has larger spatial fluctuations of droplet spectral properties. A significantly larger mean spectral width in the bin microphysics across the entire cloud depth is the largest difference between the two schemes. A fourfold increase of the number of superdroplets per grid volume and a twofold increase of the spectral resolution and thus the number of bins have small impact on the results and provide only minor changes to the comparison between simulated cloud properties.
The research program for the first year of this project (see the original research proposal) consists of developing an explicit marching scheme for solving the parabolized stability equations (PSE). Performing mathematical analysis of the computational algorithm including numerical stability analysis and the determination of the proper boundary conditions needed at the boundary of the computation domain are implicit in the task. Before one can solve the parabolized stability equations for high-speed mixing layers, the mean flow must first be found. In the past, instability analysis of high-speed mixing layer has mostly been performed on mean flow profiles calculated by the boundary layer equations. In carrying out this project, it is believed that the boundary layer equations might not give an accurate enough nonparallel, nonlinear mean flow needed for parabolized stability analysis. A more accurate mean flow can, however, be found by solving the parabolized Navier-Stokes equations. The advantage of the parabolized Navier-Stokes equations is that its accuracy is consistent with the PSE method. Furthermore, the method of solution is similar. Hence, the major part of the effort of the work of this year has been devoted to the development of an explicit numerical marching scheme for the solution of the Parabolized Navier-Stokes equation as applied to the high-seed mixing layer problem.
A time-independent convection diffusion equation is studied by means of a compact finite difference scheme and numerical solutions are compared to the analytic inviscid solutions. The correct internal and external boundary layer behavior is observed, due to an inherent feature of the scheme which automatically produces upwind differencing in inviscid regions and the correct viscous behavior in viscous regions.
A time-independent convection diffusin equation is studied by means of a compact finite difference scheme and numerical solutions are compared to the analytic inviscid solutions. The correct internal and external boundary layer behavior is observed, due to an inherent feature of the scheme which automatically produces upwind differencing in inviscid regions and the correct viscous behavior in viscous regions.
Two general purpose numerical integration schemes were built into the NASA-JSC computer system. The state-of-the-art of numerical integration, the particular integrators built into the JSC computer system, and the use of the new integration packages are described. Background information about numerical integration and the variable-order, variable-stepsize Adams numerical integration technique is discussed. Results concerning the PEACE parameter optimization program are given along with recommendations and conclusions.
We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the third-order scheme: one scheme that is based on a genuinely two-dimensional Central WENO reconstruction, and another scheme that is based on a simpler dimension-by-dimension reconstruction. The simpler dimension-by-dimension variant is then extended to a multi-dimensional fifth-order scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes.
Paper describes a numerical calculation scheme for tangential slot injection (wall-wake) flows; application of the scheme over a wide range of flow conditions indicates increased accuracy compared to previous work. Predictions from the numerical code were in good agreement with experiment (velocity profile, skin-friction, and effectiveness data) for low- and high-speed flows. To achieve improved accuracy, modifications in the turbulence modeling, compared to previous research, were necessary for the imbedded shear layer region in the near field and for the wall region near shear layer impingement. Anomalous behavior was noted for far field experimental velocity profiles in low-speed flow when the slot-to-free stream velocity ratio was near one