Applied Numerical Mathematics最新文献

In this paper, we consider the numerical approximation of incompressible magnetohydrodynamic (MHD) system with variable density. Firstly, we provide first- and second-order time discretization schemes based on the convective form of the Gauge-Uzawa method. Secondly, we prove that the proposed schemes are unconditionally stable. We also provide error estimates through rigorous theoretical analysis. Then, we construct a fully-discrete first-order scheme with finite elements in space and provide its stability result. Finally, we present some numerical experiments to validate the effectiveness of the proposed schemes. Furthermore, we also present the conserved scheme and its numerical results.

In this work, we investigate a linear second-order numerical method for the Allen-Cahn equation with general mobility. The proposed scheme is a combination of the two-step first- and second-order backward differentiation formulas for time approximation and the central finite difference for spatial discretization, two additional stabilizing terms are also included. The discrete maximum bound principle of the numerical scheme is rigorously proved under mild constraints on the adjacent time-step ratio and the two stabilization parameters. Furthermore, the error estimates in $H^1$-norm for the case of constant mobility and $L^{\infty }$-norm for the general mobility case, as well as the energy stability for both cases are obtained. Finally, we present extensive numerical experiments to validate the theoretical results, and develop an adaptive time-stepping strategy to demonstrate the performance of the proposed method.

In this work we solve initial-boundary value problems associated to coupled 2D parabolic singularly perturbed systems of convection-diffusion type. The analysis is focused on the cases where the diffusion parameters are small, distinct and also they may have different order of magnitude. In such cases, overlapping regular boundary layers appear at the outflow boundary of the spatial domain. The fully discrete scheme combines the classical upwind scheme defined on an appropriate Shishkin mesh to discretize the spatial variables, and the fractional implicit Euler method joins to a decomposition of the difference operator in directions and components to integrate in time. We prove that the resulting method is uniformly convergent of first order in time and of almost first order in space. Moreover, as only small tridiagonal linear systems must be solved to advance in time, the computational cost of our method is remarkably smaller than the corresponding ones to other implicit methods considered in the previous literature for the same type of problems. The numerical results, obtained for some test problems, corroborate in practice the good behavior and the advantages of the algorithm.

In this manuscript, we propose a new family of stabilized explicit parallelizable peer methods for the solution of stiff Initial Value Problems (IVPs). These methods are derived through the employment of a class of preconditioners proposed by Bassenne et al. (2021) [5] for the construction of a family of linearly implicit Runge-Kutta (RK) schemes.

In this paper, we combine the mentioned preconditioners with explicit two-step peer methods, obtaining a new class of linearly implicit numerical schemes that admit parallelism on the stages. Through an in-depth theoretical investigation, we set free parameters of both the preconditioners and the underlying explicit methods that allow deriving new peer schemes of order two, three and four, with good stability properties and small Local Truncation Error (LTE). Numerical experiments conducted on Partial Differential Equations (PDEs) arising from application contexts show the efficiency of the new peer methods proposed here, and highlight their competitiveness with other linearly implicit numerical schemes.

In this manuscript, we propose a stable algorithm for computing the zeros of Althammer polynomials. These polynomials are orthogonal with respect to a Sobolev inner product, and are even if their degree is even, odd otherwise. Furthermore, their zeros are real, distinct, and located inside the interval $(-1,1)$. The Althammer polynomial $p_n\left(x\right)$ of degree n satisfies a long recurrence relation, whose coefficients can be arranged into a Hessenberg matrix of order n, with eigenvalues equal to the zeros of the considered polynomial.

Unfortunately, the eigenvalues of this Hessenberg matrix are very ill–conditioned, and standard balancing procedures do not improve their condition numbers. Here, we introduce a novel algorithm for computing the zeros of $p_n\left(x\right)$, which first transforms the Hessenberg matrix into a similar symmetric tridiagonal one, i.e., a matrix whose eigenvalues are perfectly conditioned, and then computes the zeros of $p_n\left(x\right)$ as the eigenvalues of the latter tridiagonal matrix. Moreover, we propose a second algorithm, faster but less accurate than the former one, which computes the zeros of $p_n\left(x\right)$ as the eigenvalues of a truncated Hessenberg matrix, obtained by properly neglecting some diagonals in the upper part of the original matrix. The computational complexity of the proposed algorithms are, respectively, $O\left(\frac{n^3}{6}\right)$, and $O\left({\ell }^{2}n\right)$, with $\ell \ll n$ in general.

In this paper, we investigate optimal control problems in heterogeneous porous media. The optimal control problem is governed by the Darcy's flow equation; where the pressure is the state variable and the source/sink is the control variable. Then we introduce the reduced optimal control problem which contains only the state variable by replacing the control variable with a dependent quantity of the state variable based on the Darcy's equation. Here we employ $C^0$ interior penalty finite element methods for the spatial discretization to solve the reduced optimal control problem resulting in a fourth-order variational inequality. We use $P_2$ Lagrange finite elements for $C^0$ interior penalty methods, which require fewer degrees of freedom than $C^1$ finite element methods. We provide a priori error estimates and stability analyses by considering a heterogeneous permeability coefficient. Several numerical examples validate the given theories and illustrate the capabilities of the proposed algorithm.

The paper proposed a second-order steady-state-preserving nonstaggered central scheme for solving one-layer and two-layer open channel flows via the flux globalization. The global flux transforms the model into the homogeneous form, avoiding the complex discretization of the source terms. However, when the traditional appropriate quadrature rule discrete the global variables, the scheme tends to maintain only the moving-water equilibrium but not the “lake at rest” equilibrium. This paper proposes a new discretization method, the steady-state discretization (SSD) method of global variables, so that not only the still-water equilibrium can be maintained, but also the moving-water equilibrium, i.e., the discharge, the energy and the global flux are equilibrium. The scheme also ensures that the cross-sectional wet area is positive by introducing a “draining” time-step technique. Numerical experiments verify that the scheme is well-balanced, positivity-preserving and robust when flowing through open channel flows under the continuous or discontinuous bottom topography and channel width, and exactly capturing small perturbations and propagating interfaces of the steady-state solution.

The solution of the singular perturbation problems (SPP) of convection-diffusion-reaction type may exhibit regular and corner layers in a rectangular domain. In this work, we construct and analyze a parameter-uniform operator-splitting alternating direction implicit (ADI) scheme to efficiently solve a two-dimensional parabolic singularly perturbed problem with two positive parameters. The proposed model is a combination of the backward-Euler method defined on a uniform mesh in time and a hybrid method in space defined on a special Shishkin mesh. The analysis is presented on a layer adapted piecewise-uniform Shishkin mesh. The developed numerical method is proved to be first-order convergent in time and almost second-order convergent in space. The numerical experiments are performed to validate the theoretical convergence results and illustrate the efficiency of the current strategy.

In recent years, there has been a growing interest in leveraging Scientific Machine Learning (SciML) techniques to address challenges in solving Partial Differential Equations (PDEs). This study focuses on forecasting the growth of microbial populations in soil using a novel numerical methodology, the Physics-Informed Neural Network (PINN). This approach is crucial in overcoming the inherent challenges associated with the general unculturability of soil bacteria. PINNs can be used to model the growth of bacterial and fungal populations, considering environmental factors like temperature, solar radiation, air humidity, soil hydration status, and external weather conditions. In this paper, some stability issues related to the mathematical model have been analyzed. Moreover, by utilizing field data and applying equations that describe the biological mechanisms of microbial growth, a PINN was trained to predict the development of the microbiota over time. The results demonstrate that the use of PINNs for studying microbial growth and evolution is a promising tool for enhancing agriculture, optimizing cultivation processes, and facilitating efficient resource management.