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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Exponential Runge-Kutta Parareal for non-diffusive equations

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. Here, this paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Hyperbolic Prismatic Grid Generation and Solution of Euler Equations on Prismatic Grids

A hyperbolic grid generation method is used to generate prismatic grids and an approach using prismatic grids to solve the Euler equations is presented. The theory of the stability and feasibility of the hyperbolic grid generation method is presented. The hyperbolic grid generation method of Steger et al for structured grids is applied to a three dimensional triangularized surface definition to generate a grid that is unstructured on each successive layer. The grid, however, retains structure in the body-normal direction and has a computational cell shaped like a triangular prism. In order to take advantage of the structure in the normal direction, a finite-volume scheme that treats the unknowns along the normal direction implicitly is introduced and the flow over a sphere is simulated.

Pandya, S. A.↗
NASA NTRS1992

Fast wavelet based algorithms for linear evolution equations

A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.

Engquist, Bjorn↗
NASA NTRS1989

On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Vandommelen, Leon L.↗
NASA NTRS1990

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Van Dommelen, Leon L.↗
NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗
NASA NTRS1973

Numerical calculation of flow fields about rectangular wings of finite thickness in supersonic flow

The calculation of the outer inviscid flow about a rectangular wing moving at supersonic speeds is reported. The inviscid equations of motion governing the flow generated by the wing form a set of hyperbolic differential equations. The flow field about the rectangular wing is separated into three regions consisting of the forebody, the afterbody, and the wing wake. Solutions for the forebody are obtained using conical flow techniques while the afterbody and the wing wake regions are treated as initial value problems. The numerical solutions are compared in the two dimensional regions with known exact solutions.

Vogel, J. M.↗
NASA NTRS1986

Regularity estimates up to the boundary for elliptic systems of difference equations

Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

Strikwerda, J. C.↗
NASA NTRS1988

Calculation of non-stationary aerodynamic forces in the near sonic range

The development of a mathematical method for calculating nonstationary supersonic flow in the near-sonic range is described. A perturbation formula is derived based on the exact stationary values; it is applicable to the equation for potential. The problem can thus be divided into stationary and nonstationary fields. The pressure distribution in an oscillating profile is determined, based on hyperbolic differential equations. It is shown that there are important corollaries concerning the application of linear theory. With suitable extrapolations, linear theory can be used up to about Mach 0.8. Linear theory is not applicable, however, when determining the moment coefficients; for this case, a special technique is described.

Teipel, I.↗
DOE OSTI2021

Development, Implementation, and Verification of Partially-Ionized Collisional Multifluid Plasma Models in Drekar

We consider the development of multifluid models for partially ionized multispecies plasmas. The models are composed of a standard set of five-moment fluid equations for each species plus a description of electromagnetics. The most general model considered utilizes a full set of fluid equations for each charge state of each atomic species, plus a set of fluid equations for electrons. The fluid equations are coupled through source terms describing electromagnetic coupling, ionization, recombination, charge exchange, and elastic scattering collisions in the low-density coronal limit. The form of each of these source terms is described in detail, and references for required rate coefficients are identified for a diverse range of atomic species. Initial efforts have been made to extend these models to incorporate some higher-density collisional effects, including ionization potential depression and three- body recombination. Some reductions of the general multifluid model are considered. First, a reduced multifluid model is derived which averages over all of the charge states (including neutrals) of each atomic species in the general multifluid model. The resulting model maintains full consistency with the general multifluid model from which it is derived by leveraging a quasi-steady-state collisional ionization equilibrium assumption to recover the ionization fractions required to make use of the general collision models. Further reductions are briefly considered to derive certain components of a single-fluid magnetohydrodynamics (MHD) model. In this case, a generalized Ohm's law is obtained, and the standard MHD resistivity is expressed in terms of the collisional models used in the general multifluid model. A number of numerical considerations required to obtain robust implementations of these multifluid models are discussed. First, an algebraic flux correction (AFC) stabilization approach for a continuous Galerkin finite element discretization of the multifluid system is described in which the characteristic speeds used in the stabilization of the fluid systems are synchronized across all species in the model. It is demonstrated that this synchronization is crucial in order to obtain a robust discretization of the multifluid system. Additionally, several different formulations are considered for describing the electromagnetics portion of the multifluid system using nodal continuous Galerkin finite element discretizations. The formulations considered include a parabolic divergence cleaning method and an implicit projection method for the traditional curl formulation of Maxwell's equations, a purely- hyperbolic potential-based formulation of Maxwell's equations, and a mixed hyperbolic-elliptic potential-based formulation of Maxwell's equations. Some advantages and disadvantages of each formulation are explored to compare solution robustness and the ease of use of each formulation. Numerical results are presented to demonstrate the accuracy and robustness of various components of our implementation. Analytic solutions for a spatially homogeneous damped plasma oscillation are derived in order to verify the implementation of the source terms for electromagnetic coupling and elastic collisions between fluid species. Ionization balance as a function of electron temperature is evaluated for several atomic species of interest by comparing to steady-state calculations using various sets of ionization and recombination rate coefficients. Several test problems in one and two spatial dimensions are used to demonstrate the accuracy and robustness of the discretization and stabilization approach for the fluid components of the multifluid system. This includes standard test problems for electrostatic and electromagnetic shock tubes in the two-fluid and ideal shock-MHD limits, a cylindrical diocotron instability, and the GEM challenge magnetic reconnection problem. A one-dimensional simplified prototype of an argon gas puff configuration as deployed on Sandia's Z-machine is used as a demonstration to exercise the full range of capabilities associated with the general multifluid model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2012

Assessing Tsunami Vulnerabilities of Geographies with Shallow Water Equations

Tsunami preparedness is crucial for saving human lives in case of disasters that involve massive water movement. In this work, we develop a framework for visual assessment of tsunami preparedness of geographies. Shallow water equations (also called Saint Venant equations) are a set of hyperbolic partial differential equations that are derived by depth-integrating the Navier-Stokes equations and provide a great abstraction of water masses that have lower depths compared to their free surface area. Our specific contribution in this study is to use Microsoft's XNA Game Studio to import underwater and shore line geographies, create different tsunami scenarios, and visualize the propagation of the waves and their impact on the shore line geography. Most importantly, we utilized the computational power of graphical processing units (GPUs) as HLSL based shader files and delegated all of the heavy computations to the GPU. Finally, we also conducted a validation study, in which we have tested our model against a controlled shallow water experiment. We believe that such a framework with an easy to use interface that is based on readily available software libraries, which are widely available and easily distributable, would encourage not only researchers, but also educators to showcase ideas.

Aras, Rifat↗
DOE OSTI2020

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

This study presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time-stepping operator, analyzed directly and bounded in the norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time-stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge--Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge--Kutta schemes.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Time-Dependent Parabolic Finite Difference Formulation for Harmonic Sound Propagation in a Two-Dimensional Duct with Flow

An explicit finite difference real time iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for future large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable for a harmonic monochromatic sound field, a parabolic (in time) approximation is introduced to reduce the order of the governing equation. The analysis begins with a harmonic sound source radiating into a quiescent duct. This fully explicit iteration method then calculates stepwise in time to obtain the 'steady state' harmonic solutions of the acoustic field. For stability, applications of conventional impedance boundary conditions requires coupling to explicit hyperbolic difference equations at the boundary. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Kreider, Kevin L.↗
NASA NTRS2001

Numerical Simulation of Two Phase Flows

Two phase flows can be found in broad situations in nature, biology, and industry devices and can involve diverse and complex mechanisms. While the physical models may be specific for certain situations, the mathematical formulation and numerical treatment for solving the governing equations can be general. Hence, we will require information concerning each individual phase as needed in a single phase. but also the interactions between them. These interaction terms, however, pose additional numerical challenges because they are beyond the basis that we use to construct modern numerical schemes, namely the hyperbolicity of equations. Moreover, due to disparate differences in time scales, fluid compressibility and nonlinearity become acute, further complicating the numerical procedures. In this paper, we will show the ideas and procedure how the AUSM-family schemes are extended for solving two phase flows problems. Specifically, both phases are assumed in thermodynamic equilibrium, namely, the time scales involved in phase interactions are extremely short in comparison with those in fluid speeds and pressure fluctuations. Details of the numerical formulation and issues involved are discussed and the effectiveness of the method are demonstrated for several industrial examples.

Liou, Meng-Sing↗
NASA NTRS1988

Explicit upwind algorithm for the parabolized Navier-Stokes equations

A new explicit upwind algorithm based on Roe's flux-difference splitting (FDS) method has been developed for the three-dimensional Parabolized Navier-Stokes (PNS) equations. For three-dimensional flows, FDS's are determined separately for the two nonmarching directions and modified to account for the calculated shock angle in the crossflow plane. Second-order FDS is applied to the pressure and convection terms with the streamwise pressure gradient limited in the subsonic region to maintain a hyperbolic inviscid equation set. Second-order central differencing is obtained in the two-step algorithm for the shear and heat flux terms. The new algorithm is demonstrated for three laminar flow test cases: supersonic flow over a flat plate, hypersonic flow over a 15 deg ramp, and hypersonic flow past a 10 deg cone at a 24 deg angle of attack. The computed results agree well with experimental measurements.

Korte, John J.↗