Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

An improved third-order HWCNS for compressible flow simulation on curvilinear grids

Due to the very high requirements on the quality of computational grids, stability property and computational efficiency, the application of high-order schemes to complex flow simulation is greatly constrained. In order to solve these problems, the third-order hybrid cell-edge and cell-node weighted compact nonlinear scheme (HWCNS3) is improved by introducing a new nonlinear weighting mechanism. The new scheme uses only the central stencil to reconstruct the cell boundary value, which makes the convergence of the scheme more stable. The application of the scheme to Euler equations on curvilinear grids is also discussed. Numerical results show that the new HWCNS3 achieves the expected order in smooth regions, captures discontinuities sharply without obvious oscillation, has higher resolution than the original one and preserves freestream and vortex on curvilinear grids.

Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction

This paper studies the convergence properties of an arbitrary Lagrangian–Eulerian (ALE) Riemann-based SPH algorithm in conjunction with a Weighted Essentially Non-Oscillatory (WENO) high-order spatial reconstruction, in the framework of the DualSPHysics open-source code. A convergence analysis is carried out for Lagrangian and Eulerian simulations and the numerical results demonstrate that, in absence of particle disorder, the overall convergence of the scheme is close to the one guaranteed by the WENO spatial reconstruction. Moreover, an alternative method for the WENO spatial reconstruction is introduced which guarantees a speed-up of 3.5, in comparison with the classical Moving Least-Squares (MLS) approach.

An Improved Third-order HWCNS for Compressible Flow Simulation on Curvilinear Grids

Due to the very high requirements on the quality of computational grids, stability property and computational efficiency, the application of high-order schemes to complex flow simulation is greatly constrained. In order to solve these problems, the third-order hybrid cell-edge and cell-node weighted compact nonlinear scheme(HWCNS3) is improved by introducing a new nonlinear weighting mechanism. The new scheme uses only the central stencil to reconstruct the cell boundary value, which makes the convergence of the scheme more stable. The application of the scheme to Euler equation on curvilinear grids is also discussed. Numerical results show that the new HWCNS3 achieves the expected order in smooth region, captures discontinuities sharply without obvious oscillation, has higher resolution than the original one and preserves freestream and vortex on curvilinear grids.

Numerical Simulation for a Fractal MIM Model for Solute Transport in Porous Media

A fractal mobile-immobile (MIM in short) model for solute transport in heterogeneous porous media is investigated from numerics. An implicit finite difference scheme is set forth for solving the coupled system, and stability and convergence of the scheme are proved based on the estimate of the spectral radius of the coefficient matrix. Numerical simulations with different parameters are presented to reveal the solute transport behaviors in the fractal case.

Mass-conserving diffusion-based dynamics on graphs

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

Modified Dai-Yuan iterative scheme for Nonlinear Systems and its Application

<p style='text-indent:20px;'>By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.</p>

Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.