This paper establishes the sufficient criteria for the stability in distribution of numerical solutions for the stochastic functional differential equation (SFDE) with infinite delay. We first check the mean-square boundedness and convergence of numerical solutions from different initial data and prove the existence of the invariant measure of the approximate segment process under suitable conditions. Then, we show that the numerical stationary distribution converges to the stationary distribution of the underlying solution.

This work focuses on the construction and numerical analysis of novel time-filtered second-derivative methods for stiff equations. The procedure is based on the recent first-derivative time filters (DeCaria et al., 2022 [6]). Such methods are developed by incorporating inexpensive pre-filtering and post-filtering steps into existing second-derivative schemes. We show that applying these filtering steps to second-derivative multistep or multi-stage methods results in new methods that combine features of both multistep and multi-stage approaches, referred to as second-derivative general linear methods (SGLMs). The well-established properties of SGLMs are utilized to analyze the accuracy and stability of the filtered methods and to design optimal new filters for time-stepping schemes. Several new embedded families of high-accuracy methods with low cognitive complexity and excellent stability characteristics are introduced. Finally, numerical experiments validate the stability and efficiency of the proposed methods.

Nonsingular H-matrices play an important role in economic mathematics and cybernetics. In this paper, by using the scaling technique for inequalities and constructing a positive diagonal matrix, some new criteria for nonsingular H-matrices are derived. Additionally, a necessary condition for identifying nonsingular H-matrices is presented. Finally, some numerical examples are provided to illustrate the validity of our results.

This article explores the existence and controllability results within the classes of second-order functional integro-differential equations with infinite delay. The study uses a Schauder fixed-point theorem that works in a stochastic domain to show that there are mild solutions to this problem. Additionally, rigorous proofs outlining the controllability of these problems despite the complexities introduced by infinite delay are provided. As an example of how the theory can be used in real life, an example is given to show that the established theory works for dealing with systems with infinite delay in a controlled environment.

Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic systems with non-conservative products almost as efficiently as conservation laws. However, some PDE systems, like computational electrodynamics (CED), magnetohydrodynamics (MHD), and relativistic magnetohydrodynamics (RMHD), have involution constraints that require divergence-free or divergence-preserving evolution of vector fields. In such situations, a Yee-style collocation of variables proves indispensable, and that collocation is retained in this work. In previous works, only higher order finite volume (FV) discretization of such involution-constrained systems was possible. In this work, we show that substantially more efficient AFD-WENO methods have been extended to encompass divergence-preserving hyperbolic PDEs. Our method retains the Yee-style collocation of normal components of the divergence-free/preserving vector field. However, the variables that require zone-centered evolution are evolved with AFD-WENO methods. Since those variables make up the bulk of the primal variables for the PDE of interest, this results in substantial savings in computational complexity. Even the volumetric reconstruction of the divergence-free/preserving vector field is bypassed. Instead, we realize that any divergence-preserving update of a vector field must have a general form. This general form looks closely like the familiar induction equation that is well known in CED or MHD. We exploit the generality of that form to extract the edge-centered variables that are needed in the update of the facially averaged vector field components. The two-dimensional (2D) Riemann solver is used to provide us with multidimensionally stabilized versions of these update terms. The generality of our approach is demonstrated by the fact that problems in CED, MHD, and RMHD can all be solved by the same general AFD-WENO algorithm that is presented here. Spatial accuracies up to the ninth order of accuracy are demonstrated. Several stringent test problems from CED, MHD, and RMHD are shown. We also show that the algorithm takes well to the physical constraint-preserving (PCP) formulation of AFD-WENO schemes that was presented by the authors. The efficient and time-explicit PCP strategy for divergence-preserving PDEs that we have presented here extends the applicability of our method to very stringent MHD and RMHD problems.

Several important physical phenomena in engineering and scientific fields are modeled by non-local fractional models. The development of numerical schemes becomes crucial due to the non-availability of the exact solutions to such models. However, the numerical approximation of these models is challenging and imposes several computational constraints. In this paper, we have devised two highly efficient, energy-stable numerical schemes to solve space fractional reaction-diffusion models. Spatial discretization is performed using a fourth-order matrix transform technique having the advantage of straightforward extension to two and higher spatial dimensions. The time-stepping schemes are developed using an exponential time differencing approach based on a third-order real-pole restricted Padé approximation and a third-order Padé(1,2) approximation. Computationally efficient versions of the schemes are constructed using a splitting technique. Algorithms based on these schemes are constructed, allowing easy coding, and implemented to perform several numerical experiments on problems of practical interest such as the enzyme kinetics equation, Fisher’s equation, and the Allen-Cahn equation. The proposed schemes allow the accurate and efficient simulation of these dynamical models. Solution profiles are plotted to demonstrate the effectiveness of these schemes. Convergence results are computed to validate the accuracy, and central processing unit time is recorded to show the computational efficiency.

The fractional Laplacian has been extensively applied to model nonlocal phenomena such as anomalous diffusion. In this paper, we propose to employ the Gegenbauer collocation method to discretize one special case of the fractional Laplacian-Riesz fractional derivative of variable-order (VO). We analyze the features of the collocation method for both constant-order and VO cases and develop an efficient preconditioner. We present some approximation errors and several numerical tests to confirm the efficiency of the method.

This paper investigates the strong convergence of a numerical method for stochastic subdiffusion, which involves a Caputo derivative in time of order [inline-graphic not available: see fulltext], and is driven by fractionally integrated multiplicative space-time white noise. A spatio-temporal full discretization method, based on the fractional exponential integrator scheme and the finite element method, is constructed and analyzed. The error estimation is quite complex due to the following inherent difficulties: (i) the solution operator corresponding to the mild solution does not satisfy the semigroup property because of the memory effects induced by the fractional derivative, (ii) the space-time white noise is coarser than the general trace class noise, and the regularity of the solution is very low. The well-posedness of the underlying equation is established via the equivalent norm, and a sharp strong convergence rate of the fully discrete solution is derived using Mainardi’s Wright function. Finally, numerical experiments are presented to validate the theoretical results.

The initial value problem (IVP) solutions of two (3+2)-dimensional integrable nonlinear equations, such as generalized Sawada-Kotera (SK) and generalized Date-Jimbo-Kashiwara-Miwa (DJKM) equations, which are constructed by the extension of the corresponding two-dimensional equations, are solved by means of the non-local ${\bar{\partial }}$-method. The above significant results provide a good inspiration for dealing with similar high-dimensional nonlinear equations.

This paper is concerned with entropy solutions of scalar conservation laws of the form

where the flux $ f = f(x, u) $ depends explicitly on the spatial variable $ x $. Using an extension of Kruzkov’s doubling variable method, we establish contraction properties of entropy solutions under minimal regularity assumptions on the flux, as well as the uniqueness of entropy solutions. The flux is assumed to be locally Lipschitz, along with some additional conditions.

The class of SDD$_1$ matrices is an important subclass of H-matrices. The strictly diagonally dominant (SDD) matrices, the doubly strictly diagonally dominant (DSDD) matrices, and the Dashnic Zusmanovich type (DZT) matrices are included in SDD$_1$ matrices. This paper presents an infinity norm upper bound for the inverse of SDD$_1$ matrices using the definition of the matrix norm. We prove that it is sharper than the well-known Varah’s bound for SDD matrices, and it generally performs better than the existing bounds for SDD$_1$, DSDD, and DZT matrices. As an application, an error bound for the linear complementarity problems (LCPs) of B$_1$-matrices is given.

The focal point of this paper is to theoretically investigate and numerically validate the effect of time delay on the exponential stabilization of a class of coupled hyperbolic systems with a delay term in the internal feedback. The class in question consists of two strongly coupled wave equations featuring delayed and non-delayed damping terms on the first wave equation. Through a standard change of variables and semigroup theory, we address the well-posedness of the considered coupled system. Thereon, based on some observability inequalities, we derive sufficient conditions guaranteeing the exponential decay of a suitable energy. On the other hand, from the numerical point of view, we validate the theoretical results in one-dimensional domains based on a suitable numerical approximation obtained through the Finite Difference Method. More precisely, we construct a discrete numerical scheme, which in addition to being converging and conditionally stable, preserves the energy decay property of its continuous counterpart. The theoretical analysis and the implementation of our developed numerical scheme assert the effect of the time-delayed damping on the exponential stability of strongly coupled wave equations.

In this paper, a new finite-volume multi-resolution weighted essentially non-oscillatory (MR-WENO) scheme is developed on mixed-element unstructured meshes. The reconstruction process is simplified to avoid the loss of the accuracy by using the polynomial on each stencil instead of their linear combinations in defining the smoothness indicator. Therefore, the modified MR-WENO scheme could obtain the optimal order of the accuracy in smooth regions with arbitrary positive linear weights. Moreover, two selection methods of the stencils are given based on edge neighbors and vertex neighbors, respectively; it is found that the vertex-based selection method performs well due to the compactness. Several numerical examples are given to illustrate that the new MR-WENO scheme has good performance on mixed-element meshes.

In this work, we develop a novel sparsity reconstruction algorithm for Electrical Impedance Tomography (EIT), focusing on problems where the region of interest contains relatively simple, localized inhomogeneities. The optimization framework employs $l^1$-norm regularization to promote sparsity, combined with the soft shrinkage algorithm. Our key ingredient is incorporating the Adam (Adaptive Moment Estimation) algorithm for adaptive step sizes and directions, enhancing both convergence speed and accuracy. The optimization problem is solved numerically using a high-order Discontinuous Galerkin (DG) method with quadratic polynomials. Numerical results demonstrate the efficiency and accuracy of the proposed sparsity reconstruction method with Adam, with comparisons to the smoothness regularization also presented.

We investigate the application of an analytically divergence-free kernel method for solving the Stokes equations on bounded domains in this paper. We develop a new collocation method based on the divergence-free kernel trial spaces produced by radial basis functions (RBFs). An $L^{2}$ error is obtained when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solutions, the smoothness of the computational domain, the selection of the scaling parameter, and the approximation of divergence-free kernel trial spaces (for the velocity field) and RBF trial spaces (for the pressure field). The convergence theory is established in Sobolev spaces, which covers stationary and non-stationary approximations. Several numerical examples are provided to demonstrate numerical efficiency.

In this paper, we developed novel fourth-order Runge-Kutta type exponential time differencing (ETD) A-stable and L-stable methods for space-fractional nonlinear reaction-diffusion equations with initial non-smooth or smooth data. Based on compact finite differences, a fourth-order technique is used for spatial discretization, while ETD is employed to discretize the time. Our novel numerical schemes have the benefit of explicitly handling the nonlinear term. The well-known issue of numerical instability related to computing the matrix exponential is addressed using the real single-pole rational approximation, namely the restricted Padé approximation approach. The corresponding ETD-A-stable and L-stable methods are obtained. Convergence, error estimates, and stability analysis of the suggested approaches are studied theoretically. Under a global Lipschitz continuity assumption, the unconditional $L^2$ numerical stability is established. Moreover, the convergence order of ${\mathcal {O}}(k^4)$ for the derived methods is also studied in the norm $L^2$. Numerical experiments demonstrate the advantages of the methods in computational accuracy, efficiency, and reliability.