Journal of Scientific Computing最新文献

Starting from Kirchhoff-Huygens representation and Duhamel’s principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard’s ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using Fourier transform in time, we derive the corresponding Eulerian short-time propagator in the frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose a time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green’s functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement the TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.

Recently, tensor decompositions have attracted increasing attention and shown promising performance in processing multi-dimensional data. However, the existing tensor decompositions assume that the correlation along one mode is homogeneous and thus cannot characterize the multiple types of correlations (i.e., heterogeneous correlation) along the mode in real data. To address this issue, we propose a heterogeneous tensor product that allows us to explore this heterogeneous correlation, which can degenerate into the classic tensor products (e.g., mode product and tensor–tensor product). Equipped with this heterogeneous tensor product, we develop a generalized tensor decomposition (GTD) framework for third-order tensors, which not only induces many novel tensor decompositions but also helps us to better understand the interrelationships between the new tensor decompositions and the existing tensor decompositions. Especially, under the GTD framework, we find that new tensor decompositions can faithfully characterize the multiple types of correlations along the mode. To examine the effectiveness of the new tensor decomposition, we evaluate its performance on a representative data compression task. Extensive experimental results on multispectral images, light field images, and videos compression demonstrate the superior performance of our developed tensor decomposition compared to other existing tensor decompositions.

In this paper, a linearized decoupled mass and energy conservation Crank-Nicolson (CN) fully-discrete scheme is proposed for the coupled nonlinear Schrödinger (CNLS) system with the conforming bilinear Galerkin finite element method (FEM), and the unconditional supercloseness and superconvergence error estimates in (H^1)-norm are deduced rigorously. Firstly, with the aid of the popular time-space splitting technique, that is, by introducing a suitable time discrete system, the error is divided into two parts, the time error and spatial error, the boundedness of numerical solution in (L^infty )-norm is derived strictly without any constraint between the mesh size h and the time step (tau ). Then, thanks to the high accuracy result between the interpolation and Ritz projection, the unconditional superclose error estimate is obtained, and the corresponding unconditional superconvergence result is acquired through the interpolation post-processing technique. At last, some numerical results are supplied to verify the theoretical analysis.

The Legendre–Gauss–Labatto spectral collocation method is proposed to solve the fully nonlinear Monge-Ampère equation in both two and three dimensional settings with the Dirichlet boundary conditions. The inhomogeneous boundary conditions are effectively handled by converting to homogeneous boundary conditions or modifying the second-order differentiation matrices. We propose a novel approach for approximating the initial value, which significantly reduces the number of iteration steps, thus simplifying the computations compared to existing methods. To overcome the strong nonlinearity of the underlying equation, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its spectral collocation approximations. The convergence analysis of the proposed scheme is discussed under (H^1)-, (H^2)- and (L^2)-norms. Numerical examples are presented to validate the theoretical estimates. Several interesting phenomena are observed for the first time and open for mathematical verification.

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

Conservation and numerical integration have been important issues for finite difference method related to robustness, reliability and accuracy requirements. In this paper, we discuss the relationship between the discretized Newton–Leibniz formula and four conservation and integration properties, including geometric conservation, flow conservation, surface integration and volume integration, for the multi-block based high-order cell-centered finite difference method. In order to achieve these conservation and integration properties, as well as multi-block compatibility, high-order accuracy, and stability within a unified methodology, we propose a new series of boundary schemes that incorporate all these constraints. To ensure geometric conservation, conservative metrics and Jacobian are adopted for coodinate transformation. To realize flow conservation, the width of the boundary stencil is enlarged to provide more degrees of freedom in order to meet the conservation constraints. To achieve uniformly high-order accuracy with arbitrary multi-block topology, cross-interface interpolation or differencing is avoided by utilizing one-sided scheme. To maintain stability, boundary interpolation scheme is designed as upwindly and compactly as possible. The proposed method is finally tested through a series of numerical cases, including a wave propagation and an isentropic vortex for accuracy verification, several acoustic tests to demonstrate the capability of handling arbitrary multi-block grid topology, a wavy channel and a closed flying wing problem for conservation verification. These numerical tests indicate that the new scheme possesses satisfactory conservation and integration properties while satisfying the requirements for high-order accuracy and stability.

We investigate an inverse problem with quasi-boundary value regularization for reconstructing a source term of time-space fractional diffusion equations from the final observation. A sine transform based preconditioning technique is developed for the linear system which arises from the finite difference discretization of the regularized problem. By making use of the special structure, the proposed preconditioner can be inverted efficiently by the fast sine transform and fast Fourier transform. Theoretically, we show that the preconditioned matrix can be written as the sum of two matrices. The eigenvalues of one matrix are located within a rectangular domain which is uniformly bounded away from zero. Moreover, the boundaries of the domain are independent of grid numbers, regularization parameter, and the noise level. The other matrix has rank less than twice the number of spatial grids but is independent of the number of temporal grids. Numerical experiments are performed to verify the effectiveness of the proposed preconditioner.

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced optimal transport approximation, built upon the Sinkhorn algorithm. We also present accelerated schemes for maximum mean discrepancies involving kernels. Our approaches reduce the arithmetic operations needed to compute distances from ({{mathcal {O}}}left( n^{2}right) ) to ({{{mathcal {O}}}}left( n log n right) ), opening the door to handle large and high-dimensional datasets efficiently. Furthermore, we establish robust connections between transportation problems, encompassing Wasserstein distance and unbalanced optimal transport, and maximum mean discrepancies. This empowers practitioners with compelling rationale to opt for adaptable distances.

A high-order time discretization scheme to approximate the time-fractional wave equation with the Caputo fractional derivative of order (alpha in (1, 2)) is studied. We establish a high-order formula for approximating the Caputo fractional derivative of order (alpha in (1, 2)). Based on this approximation, we propose a novel numerical method to solve the time-fractional wave equation. Remarkably, this method corrects only one starting step and demonstrates second-order convergence in both homogeneous and inhomogeneous cases, regardless of whether the data is smooth or nonsmooth. We also analyze the stability region associated with the proposed numerical method. Some numerical examples are given to elucidate the convergence analysis.

This paper investigates the numerical algorithm and its error estimates for the dynamics of relativistic charged particles under a strong maximal ordering scaling magnetic field. To maintain the fundamental principles of relativistic dynamics, including energy conservation, volume preservation, and the Lorentz invariant property, we construct a structure-preserving algorithm using the splitting scheme. This algorithm ensures the preservation of volume, energy, and the Lorentz invariant property (VELPA) exactly. Specifically, we establish an uniform and optimal error bound in both 4-position and 4-velocity for VELPA. Numerical experiments are also presented to demonstrate the advantages of VELPA in both uniform error estimate and conservation of energy, compared to the implicit Euler method and traditional energy-preserving AVF method.