Search NASA⌕ Search

SEARCH · Search NASA

Results for “fast solver”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Fast solvers for tokamak fluid models with PETSc

Multigrid (MG) is widely recognized as a highly effective solver for the model problem, the Laplacian, but textbook MG fails on most problems of interest. MG methods have been applied to complex, real-world applications with careful consideration of the physical model and discretization. In this work we develop the first step in applying MG methods to science and engineering relevant magnetohydrodynamics (MHD) tokamak models in the M3D-C1 (https://m3dc1.pppl.gov) fusion energy science code. The semi-implicit time integrator in M3D-C1 is composed of many linear solves. The implicit advance of the momentum equation is the most challenging and is the focus of this work. The current production solver in M3D-C1 is a block Jacobi (BJ) preconditioner within a Krylov solver, where blocks group degrees of freedom on planes of constant toroidal coordinate. BJ convergence degrades as the number of planes increases due to the spectral properties of the matrix preconditioned with BJ. The partially magnetic field-aligned, regular toroidal grid structure in M3D-C1 is amenable to semi-coarsening geometric MG in the toroidal direction. This paper develops such a solver and demonstrates competitive performance on a runaway electron model of a SPARC (https://cfs.energy/technology/sparc) disruption, and superior robustness on a stellarator model on which the BJ solver fails to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1980

A Navier-Stokes fast solver for turbulence modeling applications

A computer code for the evaluation and/or optimization of the predicative potential of second order turbulent closure models in simple two dimensional flow configurations is discussed. A procedure for the numerical solution of the steady constant property Navier-Stokes equations are described together with algebraic, one dimensional and two dimensional equations of turbulence closure models. Four turbulence models are compared with several sets of experimental data. The effects of initial conditions and boundary conditions are also described. The effects of purely numerical parameters, such as mesh size, boundary locations, and convergence criteria are presented.

Murphy, J. D.↗
NASA NTRS1995

A fast Poisson solver for unsteady incompressible Navier-Stokes equations on the half-staggered grid

In this paper, a fast Poisson solver for unsteady, incompressible Navier-Stokes equations with finite difference methods on the non-uniform, half-staggered grid is presented. To achieve this, new algorithms for diagonalizing a semi-definite pair are developed. Our fast solver can also be extended to the three dimensional case. The motivation and related issues in using this second kind of staggered grid are also discussed. Numerical testing has indicated the effectiveness of this algorithm.

Golub, G. H.↗
NASA NTRS1988

Fast Euler solver for transonic airfoils. I - Theory. II - Applications

Equations written in terms of generalized Riemann variables are presently integrated by inverting six bidiagonal matrices and two tridiagonal matrices, using an implicit Euler solver that is based on the lambda-formulation. The solution is found on a C-grid whose boundaries are very close to the airfoil. The fast solver is then applied to the computation of several flowfields on a NACA 0012 airfoil at various Mach number and alpha values, yielding results that are primarily concerned with transonic flows. The effects of grid fineness and boundary distances are analyzed; the code is found to be robust and accurate, as well as fast.

Dadone, Andrea↗
NASA NTRS1990

Multitasking domain decomposition fast Poisson solvers on the Cray Y-MP

The results of multitasking implementation of a domain decomposition fast Poisson solver on eight processors of the Cray Y-MP are presented. The object of this research is to study the performance of domain decomposition methods on a Cray supercomputer and to analyze the performance of different multitasking techniques using highly parallel algorithms. Two implementations of multitasking are considered: macrotasking (parallelism at the subroutine level) and microtasking (parallelism at the do-loop level). A conventional FFT-based fast Poisson solver is also multitasked. The results of different implementations are compared and analyzed. A speedup of over 7.4 on the Cray Y-MP running in a dedicated environment is achieved for all cases.

Chan, Tony F.↗
NASA NTRS1988

A simplified analysis of the multigrid V-cycle as a fast elliptic solver

For special model problems, Fourier analysis gives exact convergence rates for the two-grid multigrid cycle and, for more general problems, provides estimates of the two-grid convergence rates via local mode analysis. A method is presented for obtaining mutigrid convergence rate estimates for cycles involving more than two grids (using essentially the same analysis as for the two-grid cycle). For the simple cast of the V-cycle used as a fast Laplace solver on the unit square, the k-grid convergence rate bounds obtained by this method are sharper than the bounds predicted by the variational theory. Both theoretical justification and experimental evidence are presented.

Decker, Naomi H.↗
NASA NTRS1989

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR.

Gallopoulos, E.↗
NASA NTRS1989

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block tridiagonal matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconstant coefficients. A method was recently proposed to parallelize and vectorize BCR. In this paper, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational compelxity lower than that of parallel BCR.

Gallopoulos, E.↗
NASA NTRS1992

A fast direct solver for a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of two-dimensional separable elliptic equations on the sphere is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wavenumber and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗
DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Fast Euler solver for steady, 1-dimensional flows

A numerical technique to solve the Euler equations for steady, one dimensional flows is presented. The technique is essentially implicit, but is structured as a sequence of explicit solutions for each Riemann variable separately. Each solution is obtained by integrating in the direction prescribed by the propagation of the Riemann variables. The technique is second-order accurate. It requires very few steps for convergence, and each step requires a minimal number of operations. Therefore, it is three orders of magnitude more efficient than a standard time-dependent technique. The technique works very well for transonic flows and provides shock fitting with errors as small as 0.001. Results are presented for subsonic problems. Errors are evaluated by comparison with exact solutions.

Moretti, G.↗
NASA NTRS1983

A fast Euler solver for steady flows

A numerical technique to solve the Euler equations for steady, two-dimensional flows is presented. The technique extends to two-dimensional problems a formulation which was found to be extremely efficient for one-dimensional flows. Generalized Riemann variables are defined along two families of orthogonal coordinates, and integrated separately, sweeping back and forth alternatively along coordinate lines. The technique is second-order accurate and converges very rapidly. In addition, each step requires a minimal number of operations. Preliminary results for subsonic and transonic shockless flows are presented and discussed.

Moretti, G.↗
NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 1: One-step method

An algorithm for solving a large class of two- and three-dimensional nonseparable elliptic partial differential equations (PDE's) is developed and tested. It uses a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid-point dependent. It is easy to implement and applicable to a variety of boundary conditions. It is also computationally efficient, as indicated by the results of numerical comparisons with other established methods. Furthermore, the current algorithm has the advantage of possessing two important properties which the traditional iterative methods lack; that is: (1) the convergence rate is relatively insensitive to grid-cell size and aspect ratio, and (2) the convergence rate can be easily estimated by using the coefficient of the PDE being solved.

Chang, S. C.↗
NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Chang, S. C.↗
NASA NTRS1986

An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

Young, D. P.↗
NASA NTRS1986

Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor

A large class of two- and three-dimensional, nonseparable elliptic partial differential equations (PDEs) is presently solved by means of novel one-step (D'Yakanov-Gunn) and two-step (accelerated one-step) iterative procedures, using a local, discrete Fourier analysis. In addition to being easily implemented and applicable to a variety of boundary conditions, these procedures are found to be computationally efficient on the basis of the results of numerical comparison with other established methods, which lack the present one's: (1) insensitivity to grid cell size and aspect ratio, and (2) ease of convergence rate estimation by means of the coefficient of the PDE being solved. The two-step procedure is numerically demonstrated to outperform the one-step procedure in the case of PDEs with variable coefficients.

Chang, Sin-Chung↗
NASA NTRS1991

A fast upwind solver for the Euler equations on three-dimensional unstructured meshes

An upwind scheme is presented for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. Spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Higher-order differences are formed by a novel cell reconstruction process which results in computational times per cell comparable to those of structured codes. The approach yields highly resolved solutions in regions of smooth flow while avoiding oscillations across shocks without explicit limiting. Solutions are advanced in time by a 3-stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. Solutions are presented for a range of configurations in the transonic speed regime to demonstrate code accuracy, speed, and robustness. The results include an assessment of grid sensitivity and convergence acceleration by mesh sequencing.

Frink, Neal T.↗