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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

A parallel iterative solution method for systems of nonlinear hyperbolic equations

An iterative algorithm suitable for the solution of a system of nonlinear hyperbolic partial differentiation equations in multiple dimensions is discussed. Current numerical methods for systems of nonlinear PDEs have limited parallelism due to strong coupling between the equations. This method decouples the PDEs by linearizing the convention coefficient for a space-time domain. This provides large grain parallelism. The linearization also allows the treatment of some terms in the equations as source terms, providing more freedom to choose from a wider variety of numerical methods. Smaller grain parallelism may be exploited within the solves for each equation. Thus, the method has potential for parallelism at several levels.

Scroggs, Jeffrey S.↗
NASA NTRS1997

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L(sup 2)-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

Cockburn, Bernardo↗
NASA NTRS1989

An iterative method for systems of nonlinear hyperbolic equations

An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.

Scroggs, Jeffrey S.↗
NASA NTRS1990

A two-step iterative method for evolving nonlinear acoustic systems to a steady-state

A new approach for evolving two-dimensional nonlinear acoustic systems with flow to a steady state is presented. The approach is a two-step iterative method which is tested on a benchmark acoustic problem for which an exact analytical solution is available. Results are also calculated for a nonlinear acoustic problem for which an exact analytical solution is not known. Results indicate that the two-step method represents a powerful, efficient, and stable method for evolving two-dimensional acoustic systems to a steady state, and that the method is applicable to any number of spatial dimensions and to other hyperbolic systems. It is noted that for the benchmark problem only a single iteration on the method is required when the transient and steady-state field are of the same order of magnitude; however, four iterations are required when the steady-state field is several orders of magnitude smaller than the transient field. This method requires six iterations before achieving a steady state for the nonlinear test problem.

Watson, Willie R.↗
NASA NTRS2001

Discontinuous Galerkin Methods for NonLinear Differential Systems

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Barth, Timothy↗
NASA NTRS1985

A computational study of saddle point separation and horseshoe vortex system

Incompressible flow around a cylinder-end wall junction has been simulated by solving the incompressible Navier-Stokes equations in three dimensions. The equations, cast in generalized curvilinear coordinates, are solved in time as a hyperbolic system by adding a pressure term in the continuity equation and are marched to a steady state. Various physical quantities associated with the saddle point of separation and the horseshoe vortex system are calculated. Computational and experimental results are generally consistent. The skin friction and the pressure distribution on the end wall are consistent with the physics of the problem. Secondary flows both in front of the cylinder and behind it are predicted that are in qualitative agreement with flow visualization results. The calculations also indicate a strongly nonuniform pressure loading along the length of the cylinder. A new mechanism for the existence of the recirculation bubbles behind the cylinder-end wall with relatively low ratio of cylinder height to the approaching boundary layer thickness is observed which is markedly different from its two-dimensional counterpart.

Kaul, U. K.↗
NASA NTRS1986

Preconditioned methods for solving the incompressible and low speed compressible equations

Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. The compressible equations in conservation form with slow flow are also considered. Two arbitrary functions, alpha and beta, are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for beta is determined given a constant, alpha. It is further shown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which generalize previous results.

Turkel, E.↗
NASA NTRS1999

Practical Aspects of Stabilized FEM Discretizations of Nonlinear Conservation Law Systems with Convex Extension

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the POE system. Central to the development of the simplified GLS and DG methods is the Degenerative Scaling Theorem which characterizes right symmetrizes of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobean matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler, Navier-Stokes, and magnetohydrodynamic (MHD) equations. Linear and nonlinear energy stability is proven for the simplified GLS and DG methods. Spatial convergence properties of the simplified GLS and DO methods are numerical evaluated via the computation of Ringleb flow on a sequence of successively refined triangulations. Finally, we consider a posteriori error estimates for the GLS and DG demoralization assuming error functionals related to the integrated lift and drag of a body. Sample calculations in 20 are shown to validate the theory and implementation.

Barth, Timothy↗
NASA NTRS1991

Development of new flux splitting schemes

Maximizing both accuracy and efficiency has been the primary objective in designing a numerical algorithm for CFD. This is especially important for solution of complex three-dimensional systems of Navier-Stokes equations which often include turbulence modeling and chemistry effects. Recently, upwind schemes have been well received for both their capability of resolving discontinuities and their sound theoretical basis in characteristic theory for hyperbolic systems. With this in mind, two new flux splitting techniques are presented for upwind differencing.

Liou, MENG-S.↗
NASA NTRS1989

Numerical simulation of hyperbolic heat conduction with convection boundary conditions and pulse heating effects

The paper describes the numerical simulation of hyperbolic heat conduction with convection boundary conditions. The effects of a step heat loading, a sudden pulse heat loading, and an internal heat source are considered in conjunction with convection boundary conditions. Two methods of solution are presened for predicting the transient behavior of the propagating thermal disturbances. In the first method, MacCormack's predictor-corrector method is employed for integrating the hyperbolic system of equations. Next, the transfinite element method, which employs specially tailored elements, is used for accurately representing the transient response of the propagating thermal wave fronts. The agreement between the results of various numerical test cases validate the representative behavior of the thermal wave fronts. Both methods represent hyperbolic heat conduction behavior by effectively modeling the sharp discontinuities of the propagating thermal disturbances.

Glass, David E.↗
DOE OSTI2020

Interface learning of multiphysics and multiscale systems

Complex natural or engineered systems comprise multiple characteristic scales, multiple spatiotemporal domains, and even multiple physical closure laws. To address such challenges, we introduce an interface learning paradigm and put forth a data-driven closure approach based on memory embedding to provide physically correct boundary conditions at the interface. To enable the interface learning for hyperbolic systems by considering the domain of influence and wave structures into account, we put forth the concept of upwind learning toward a physics-informed domain decomposition. The promise of the proposed approach is shown for a set of canonical illustrative problems. Here, we highlight that high-performance computing environments can benefit from this methodology to reduce communication costs among processing units in emerging machine-learning-ready heterogeneous platforms toward exascale era.

42 ENGINEERING↗
NASA NTRS1998

Development of Numerical Extended Hydrodynamics for Transition-Regime Non-Equilibrium Flows Encountered in Semiconductor Manufacturing Processes

Six months of funding was received for the proposed three year research program (funding for the period from March 1, 1997 to August 31, 1997). Although the official starting date for the project was March 1, 1997, no funding for the project was received until July 1997. In the funded research period, considerable progress was made on Phase I of the proposed research program. The initial research efforts concentrated on applying the 10-, 20-, and 35-moment Gaussian-based closures to a series of standard two-dimensional non-reacting single species test flow problems, such as the flat plate, couette, channel, and rearward facing step flows, and to some other two-dimensional flows having geometries similar to those encountered in chemical-vapor deposition (CVD) reactors. Eigensystem analyses for these systems for the case of two spatial dimensions was carried out and efficient formulations of approximate Riemann solvers have been formulated using these eigenstructures. Formulations to include rotational non-equilibrium effects into the moment closure models for the treatment of polyatomic gases were explored, as the original formulations of the closure models were developed strictly for gases composed of monatomic molecules. The development of a software library and computer code for solving relaxing hyperbolic systems in two spatial dimensions of the type arising from the closure models was also initiated. The software makes use of high-resolution upwind finite-volumes schemes, multi-stage point implicit time stepping, and automatic adaptive mesh refinement (AMR) to solve the governing conservation equations for the moment closures. The initial phase of the code development was completed and a numerical investigation of the solutions of the 10-moment closure model for the simple two-dimensional test cases mentioned above was initiated. Predictions of the 10-moment model were compared to available theoretical solutions and the results of direct-simulation Monte Carlo (DSMC) calculations. The first results of this study were presented at a meeting last year.

Groth, Clinton P. T.↗
DOE OSTI2021

iharm3D: Vectorized General Relativistic Magnetohydrodynamics

iharm3D is an open-source C code for simulating black hole accretion systems in arbitrary stationary spacetimes using ideal general-relativistic magnetohydrodynamics (GRMHD). It is an implementation of the HARM (“High Accuracy Relativistic Magnetohydrodynamics”) algorithm outlined in Gammie et al. (2003) with updates as outlined in McKinney & Gammie (2004) and Noble et al. (2006). The code is most directly derived from Ryan et al. (2015) but with radiative transfer portions removed. HARM is a conservative finite-volume scheme for solving the equations of ideal GRMHD, a hyperbolic system of partial differential equations, on a logically Cartesian mesh in arbitrary coordinates.

79 ASTRONOMY AND ASTROPHYSICS↗
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↗
DOE OSTI2021

Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

Abstract An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed. The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements. The method is made invariant domain preserving for the Euler equations using convex limiting and is tested on various benchmarks.

Guermond, Jean-Luc↗
DOE OSTI2020

A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

A stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented herein. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form

We propose a one-parameter family of low-dissipation centred numerical schemes for solving hyperbolic equations in conservative or non-conservative form, using finite volume or discontinuous Galerkin finite element methods. The new schemes spring out from the multi-dimensional FORCE method and are determined by a single parameter α ≥ 1. Given an increasing sequence of real numbers 1 ≤ α 1 < α 2 < . . . < α K , there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of Courant numbers 1 > c 1 > c 2 > . . . > c K > 0 and a decreasing sequence of corresponding numerical viscosity functions d 1 > d 2 > . . . > d K . For a given Courant number c k ≤ 1 there is a real number α k > 0 and a corresponding stable scheme with minimal numerical viscosity d k . The proposed schemes suit very well the family of high-order discontinuous Galerkin finite element methods and the recently proposed class of ADER-TR schemes, whose orders of accuracy define decreasing sequences of Courant numbers, as the order of accuracy increases. The centred methods of this paper are stable in 2D and 3D in the frame of simultaneous updating formulae, unlike other centred methods, such as 1D FORCE, which are not. Furthermore, the schemes are highly accurate for slowly-moving waves, which is precisely the kind of waves that traditional centred methods smear disastrously. Additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non-conservative form. Here, the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system, for the Euler equations in one and two space dimensions, and for the Baer-Nunziato equations for compressible two-phase flow.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A method for bounding high-order finite element functions: Applications to mesh validity and bounds-preserving limiters

We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be used to locally bound any combination of these basis functions. This approach can be applied to any element/basis type at any approximation order, can provide local (i.e., subcell) extremum bounds to a desired level of accuracy, and can be evaluated efficiently on-the-fly in simulations. Furthermore, we show that this approach generally yields more accurate bounds in comparison to traditional methods based on convex hull properties (e.g., Bernstein polynomials). Furthermore, the efficacy of this technique is shown in applications such as mesh validity checks and optimization for high-order curved meshes, where positivity of the element Jacobian determinant can be ensured throughout the entire element, and continuously bounds-preserving limiters for hyperbolic systems, which can enforce maximum principle bounds across the entire solution polynomial.

Bounding box↗