Home – Home – Jornals – Numerical Analysis and Applications 2025 number 2

In the article, we consider linear second-order differential algebraic equations (DAEs) on a finite interval of integration with given initial data. A class of problems with a unique sufficiently smooth solution is identified in terms of matrix polynomials. We assume that the solution to the problem may contain rigid and rapidly oscillating components. This paper highlights the main challenges of developing algorithms for numerical solutions to the class of problems under consideration. We propose to represent the original problem in the form of a system of integral differential or integral equations with an identically degenerate matrix in front of the main part for constructing effective methods of numerical solution of second-order DAEs. Moreover, we construct numerical solution methods for problems represented in this way. These algorithms are based on explicit Adams quadrature formulas for calculating the integral term and on extrapolation formulas for other terms. The results of test example calculations are presented and analyzed.

Aiming at the problem that observation errors exist in both the observation vector and the coefficient matrix for an autoregressive model, a new parameter estimation method is proposed. First, the observation vector and coefficient matrix are recombined, which avoids the situation that the same observation value appears in both the observation vector and the coefficient matrix. Then a detailed algorithm is derived based on the principle of total least squares and indirect adjustment. Finally, the effectiveness and feasibility of the proposed method are verified by an analysis of validation and simulation examples and compared with the weighted total least squares and the correlation total least squares.

This paper introduces an approach for approximating solutions to multi-high-order fractional-differential equations by employing the Caputo fractional derivative, along with initial conditions. The technique is based on standard collocation points and Touchard polynomials. The linear equation and its initial conditions can be transformed into matrix relations by the new method, making it easier to solve a linear algebraic equation with generalized Touchard coefficients as unknowns. Also, emphasizing computational efficiency, the method is illustrated with examples.

Some iterative processes in Krylov subspaces are considered for solving systems of linear algebraic equations (SLAE) with high-order sparse matrices that arise in grid approximations of multidimensional boundary value problems. The SLAE are preconditioned by a uniform combined method that includes domain decomposition and recursive application of a two-grid algorithm, which are implemented by forming block-tridiagonal algebraic and grid structures inverted by using incomplete factorization and diagonal compensation. For some Stieltjes systems, stability and convergence of iterations are studied. Parallelization and generalization of the methods to wider classes of relevant practical problems are discussed.

Algorithms for solving ensembles of ODEs with sets of different input data arising from modeling chemical kinetics are considered with an operator splitting scheme for multiphysical calculations. The efficiency of an algorithm combining clustering of an input data ensemble and estimating of the solution within the cluster is evaluated by using a sensitivity matrix obtained by solving adjoint equations. The algorithm is implemented on the basis of numerical schemes consistent in the sense of a discrete Lagrange identity for solving ODE systems of the production-destruction type. The contribution of the clustering and the sensitivity matrix to the performance of the algorithm is evaluated. The results of testing the algorithm with atmospheric chemistry scenarios show that the algorithm allows one to reduce the calculation time with an acceptable decrease in accuracy.

We study a one-dimensional problem of an infinite tube which is open in the right and there is a piston put at one end in the left. Due to the fact that the computational domain is finite while the domain of the problem is infinite, the numerical result is affected by the presence of a reflected wave that appears when a shock travels to the right and interacts with the right boundary. Thus there is a need of implementing a non-reflecting boundary condition in order to eliminate as much as possible the effect of the reflected wave. In this paper, we use the Euler equations in mass Lagrangian coordinates as the governing equations and apply a finite volume method to compute the numerical solution. In order to deal with the reflected wave, we use a Burgers-like equation in an additional computational domain. The numerical results we obtain show that the numerical error is reduced significantly.

This paper presents a two-grid method for solving nonlinear time fractional diffusion equations (TFDEs). First, a fully discrete scheme is constructed by using P²₀- P₁ mixed finite elements (MFEs) and L1 formula for spatial and temporal discretization, respectively. Second, the stability and error of the fully discrete scheme are analyzed. Third, a two-grid algorithm (TGA) based on the fully discrete scheme is proposed and its stability and error analysis results are derived. Finally, some numerical examples are provided to support the theoretical results.