Journal of Scientific Computing最新文献

With quaternion matrices and quaternion tensors being gradually used in the color image and color video processing, the block diagonalization of block circulant quaternion matrices has become a key issue in the establishment of T-product based methods for quaternion tensors. Out of this consideration, we aim at establishing a fast calculation approach for the block diagonalization of block circulant quaternion matrices with the help of the fast Fourier transform (FFT). We first show that the discrete Fourier matrix (mathbf {F_p}) cannot diagonalize (ptimes p) circulant quaternion matrices, nor can the unitary quaternion matrices (mathbf {F_p}textbf{j}) and (mathbf {F_p}(1+textbf{j})/sqrt{2}) with (textbf{j}) being an imaginary unit of quaternion algebra. Then we prove that the unitary octonion matrix (mathbf {F_p}textbf{p}) with (textbf{p}=textbf{l},textbf{il}) or ((textbf{l}+textbf{il})/sqrt{2}) ((textbf{l}, textbf{il}) being imaginary units of octonion algebra) can diagonalize a circulant quaternion matrix of size (ptimes p), which further means that a block circulant quaternion matrix of size (mptimes np) can be block diagonalized at the cost of (O(mnplog p)) via the FFT. As one of applications, we give a fast algorithm to speed up the calculation of the T-product between (mtimes ntimes p) and (ntimes stimes p) third-order quaternion tensors via FFTs, whose computational magnitude is almost 1/p of the original one. As another application, we propose an effective compression strategy for third-order quaternion tensors with a certain low-rankness. Simulations on the color image and color video compression demonstrate that our compression strategy with no QSVD involved, can achieve higher quality compression in terms of PSNR values at much less time costs, compared with the QSVD-based methods.

Time-fractional parabolic equations with a Caputo time derivative of order (alpha in (0,1)) are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax–Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain (mtimes m) matrices (where m is the order of the collocation scheme), are verified both analytically, for all (mge 1) and all sets of collocation points, and computationally, for all ( mle 20). The semilinear case is also addressed.

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit–EXplicit Runge–Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

This paper introduces a novel approach, the mass-conservative reduced-order characteristic finite element (MCROCFE) method, designed for optimal control problem governed by convection-diffusion equation. The study delves into scenarios where the original state equation exhibits mass-conservation, yet the velocity field is non-divergence-free. The key points of emphasis are: (1) The method effectively addresses convection-dominated diffusion systems through the application of the characteristic technique; (2) Its efficiency is underscored by leveraging the proper orthogonal decomposition (POD) technique, significantly reducing the scale of solving algebraic equation systems; (3) The proposed scheme, based on the mass-conservative characteristic finite element (MCCFE) method framework and the classical POD technique with a slight adjustment to reduce-order space, maintains mass-conservation for the state variable. A priori error estimates are derived for the mass-conservative reduced-order scheme. Theoretical results are validated through numerical comparisons with the MCCFE method, emphasizing the mass-conservation, accuracy and efficiency of the MCROCFE method.

In this study, a novel preconditioner based on the absolute-value block (alpha )-circulant matrix approximation is developed, specifically designed for nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge from the numerical discretization of evolutionary equations. Our preconditioner is constructed by taking an absolute-value of a block (alpha )-circulant matrix approximation to the BLTT matrix. To apply our preconditioner, the original BLTT linear system is converted into a symmetric form by applying a time-reversing permutation transformation. Then, with our preconditioner, the preconditioned minimal residual method (MINRES) solver is employed to solve the symmetrized linear system. With properly chosen (alpha ), the eigenvalues of the preconditioned matrix are proven to be clustered around (pm 1) without any significant outliers. With the clustered spectrum, we show that the preconditioned MINRES solver for the preconditioned system has a convergence rate independent of system size. The efficacy of the proposed preconditioner is corroborated by our numerical experiments, which reveal that it attains optimal convergence.

In this paper, we present the (hbox {L}^2)-norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper.

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations (w^n_h) and (v^n_h) of (w(cdot , t_n)) and (varDelta w(cdot , t_n)) are constructed. The stability of (w^n_h) and (v^n_h) are proved, and the a priori bounds of (Vert w^n_hVert ) and (Vert v^n_hVert ) are established, remaining (alpha )-robust as (alpha rightarrow 1^{-}). Then, the error (Vert w(cdot , t_n)- w^n_hVert ) and (Vert varDelta w(cdot , t_n)-v^n_hVert ) are estimated with (alpha )-robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed (alpha ). Finally some numerical results are provided to support our theoretical findings.

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier–Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the ’90s and have several interesting features, such as being grid-free, automatically adapting to the solution by concentrating elements where the gradient is large, and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

In this paper, we formulate a model for weakly compressible two-layer shallow water flows with friction in general channels. The formulated model is non-conservative, and in contrast to the incompressible limit, our system is strictly hyperbolic. The generalized Rankine–Hugoniot conditions are provided for the present system with non-conservative products to define weak solutions. We write the Riemann invariants along each characteristic field for channels with constant width in an appendix. A robust well-balanced path-conservative semi-discrete central-upwind scheme is proposed and implemented to validate exact solutions to the Riemann problem. We also present numerical tests in general channels to show the merits of the scheme.

A new discontinuous Galerkin (DG) method is proposed and analyzed for general second-order elliptic problems. It features that local (L^2) projections are used to reconstruct the diffusion term and the convection term and that it does not need any penalty and even does not need any stabilization in the formulation. The Babus̆ka inf-sup stability is proven. The error estimates are established. More importantly, the new DG method can hold the SUPG-type stability for the convection; the SUPG-type optimal error estimates ({{mathcal {O}}}(h^{ell +1/2})) is obtained for the problem with a dominating convection for the (ell )-th order ((ell ge 0)) discontinuous element. Numerical results are provided.