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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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

A parallel algorithm for nonlinear convection-diffusion equations

A parallel algorithm for the efficient solution of nonlinear time-dependent convection-diffusion equations with small parameter on the diffusion term is presented. The method is based on a physically motivated domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. The method is suitable for the solution of problems arising in the simulation of fluid dynamics. Experimental results for a nonlinear equation in two-dimensions are presented.

Scroggs, Jeffrey S.↗
DOE OSTI2018

A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations

In this study, a Legendre spectral tau method is revisited to handle the multi-term time-fractional diffusion equations (MTT-FDEs). An error estimate and rigorous convergence analysis are carried out using some critical theorems. The applicability and accuracy of the solution method are demonstrated by a numerical example to verify the theoretical analysis.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

The Galerkin/least-squares method for advective-diffusive equations

Galerkin/least-squares finite-element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of streamline-upwind Petrov-Galerkin methods, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. Some numerical results for compressible Navier-Stokes flows are presented.

Hughes, T. J. R.↗
NASA NTRS1996

Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation

An algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail.

Abarbanel, Saul↗
DOE OSTI2024

Interface-aware Methods for Diffusion Equations [Slides]

In this talk, we will present interface-aware methods for diffusion equation. The focus is on the necessity of interface-aware, mixed cells, interface reconstruction, discontinuity of material diffusion coefficients, order of accuracy in time, property for large time steps, and simpler interface-aware methods. Numerical examples are provided to demonstrate the features on these aspects of the methods

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Applying Legendre wavelet method with Tikhonov regularization for one-dimensional time-fractional diffusion equations

In this paper, a new method based on two-dimensional Legendre wavelet basis is proposed to solve time-fractional diffusion equations. This technique is used to convert the problem into a system of linear algebraic equations via expanding the required approximation based on the Legendre wavelet basis. The effectiveness of the proposed method is examined by comparing the numerical results with other methods. When the desired system is large, it is ill-conditioned; in this case, Tikhonov regularization method, with discrepancy principle for finding the regularization parameter, is applied to stabilize the solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation

In this paper, we develop a Bernstein dual-Petrov–Galerkin method for the numerical simulation of a two-dimensional fractional diffusion equation. A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix-based approach for the spatial discretization. It is shown that the method leads to banded linear systems that can be solved efficiently. The stability and convergence of the proposed method is discussed. Finally, some numerical examples are provided to support the theoretical claims and to show the accuracy and efficiency of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Approximation rates of DeepONets for learning operators arising from advection–diffusion equations

Here we present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection–diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0 ,β 1 ) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Calculation of the neutron diffusion equation by using Homotopy Perturbation Method

The distribution of the neutrons in a nuclear fuel element in the nuclear reactor core can be calculated by the neutron diffusion theory. It is the basic and the simplest approximation for the neutron flux function in the reactor core. In this study, the neutron flux function is obtained by the Homotopy Perturbation Method (HPM) that is a new and convenient method in recent years. One-group time-independent neutron diffusion equation is examined for the most solved geometrical reactor core of spherical, cubic and cylindrical shapes, in the frame of the HPM. It is observed that the HPM produces excellent results consistent with the existing literature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

A Test Problem for Codes Solving the Discretized Diffusion Equation in Cartesian Geometry Derived Via Discrete Green’s Functions

We obtain here a solution to a zone-centered discretization of the one dimensional time-dependent diffusion equation with arbitrary initial conditions and source, constant absorption and scattering opacity, and constant zone size and time step. The solution is obtained using the discrete Green’s functions of the discretized equation. The solution of the discretized equation is useful in the testing of computer codes, because the code can be expected to agree with the solution to the discrete equation, to within small errors caused by roundoff. This is in contrast to solutions of the differential equation, with which the code results only approximately agree. Finally, the usefulness of the solution for tests of an inertial confinement fusion code is demonstrated.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Front Propagation for Reaction–Diffusion Equations in Composite Structures

We consider asymptotic problems concerning the motion of interface separating the regions of large and small values of the solution of a reaction–diffusion equation in the media consisting of domains with different characteristics (composites). Under certain conditions, the motion can be described by the Huygens principle in the appropriate Finsler (e.g., Riemannian) metric. In general, the motion of the interface has, in a sense, non-local nature. In particular, the interface may move by jumps. We are mostly concerned with the nonlinear term that is of KPP type. The results are based on limit theorems for large deviations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2000

A Two Colorable Fourth Order Compact Difference Scheme and Parallel Iterative Solution of the 3D Convection Diffusion Equation

A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it Only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with the Gauss-Seidel type iterative method. This is compared with the known 19 point fourth order compact differenCe scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and the 19 point fourth order compact schemes.

Zhang, Jun↗
NASA NTRS1995

A deterministic particle method for one-dimensional reaction-diffusion equations

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.

Mascagni, Michael↗
NASA NTRS1993

Wave and pseudo-diffusion equations from squeezed states

We show that the probability distributions P(sub n)(q,p;y) := the absolute value squared of (n(p,q;y), which are obtained from squeezed states, obey an interesting partial differential equation, to which we give two intuitive interpretations: as a wave equation in one space dimension; and as a pseudo-diffusion equation. We also study the corresponding Wehrl entropies S(sub n)(y), and we show that they have minima at zero squeezing, y = 0.

Daboul, Jamil↗
NASA NTRS1992

Effective diffusion equation in a random velocity field

The effects are studied of assumed random velocity fields on diffusion in a binary fluid. Random velocity fields can result, for example, from the high-frequency components of residual accelerations onboard spacecraft (often called g-jitter). An effective diffusion equation is derived for an average concentration which includes spatial and temporal correlations induced by the fluctuating velocity fields assumed to be Gaussianly distributed. The resulting equation becomes nonlocal, and if correlations between different components of the velocity field exist, it is also anisotropic. The simple limiting case of short correlation times is discussed and an effective diffusivity is obtained which reflects the enhanced mixing caused by the velocity fields. The results obtained in the limit of short correlation times are valid even if the probability distribution of the velocity field is not Gaussian.

Vinals, Jorge↗
DOE OSTI2018

Restoration of the Initial Data in the Problem for a Diffusion Equation with Fractional Derivative with Respect to Time

We prove the correctness of the inverse problem of finding a pair of functions: the classical solution u(x,t) of the first boundary-value problem for a linear diffusion equation with regularized fractional derivative of order α ∈ (1, 2) with respect to time in a rectangular domain (0, ℓ) × (0, t{sub 0}] and unknown initial values of the function u(x,t) for the case of additionally given values of the function at a certain fixed time t{sub 0} .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Diagonal and circulant or skew-circulant splitting preconditioners for spatial fractional diffusion equations

We propose three new preconditioners: diagonal and optimal-circulant splitting preconditioner, diagonal and skew-circulant splitting preconditioner, and diagonal and optimal-skew-circulant splitting preconditioner for solving the diagonal-plus-Toeplitz linear system discretized from the spatial fractional diffusion equations. Theoretical analysis shows that these three preconditioners can make the eigenvalues of the preconditioned matrices be clustered around 1, especially when the grids of the discretizations are refined. These results coincide with the one about the diagonal and circulant splitting preconditioner constructed recently by Bai et al. (Numer Linear Algebra Appl 24:e2093, 2017). Numerical experiments exhibit that the proposed preconditioners can significantly improve the convergence of the Krylov subspace iteration methods like GMRES and BiCGSTAB, and they outperform the diagonal and circulant splitting preconditioner as well.

97 MATHEMATICS AND COMPUTING↗