Home – Home – Jornals – Numerical Analysis and Applications 2024 number 4

Parallel algorithm for semi-implicit particle-in-cell method with energy and charge conservation The article is devoted to the construction of a parallel algorithm for calculating plasma dynamics by the particle-in-cell method using a semi-implicit scheme that conserves energy and charge. This scheme is a two-stage predictor-corrector, where at the prediction stage the semi-implicit Lapenta method is used, in which the energy-conserving linear current does not satisfy the local Gaussian law, and at the correction stage the currents, electromagnetic fields and particle velocities are corrected so that the difference laws of conservation of energy and charges were carried out accurately. This approach turns out to be effective for modeling multi-scale phenomena with a sufficiently large time step, however, it is resource-intensive, since it requires not only solving two systems of linear equations per step, but also rebuilding the entire matrix of the system. The authors have developed a matrix-operator algorithm for the software implementation of this scheme, which makes it possible to effectively parallelize calculations, as well as use various libraries for working with matrices and solvers for systems of linear equations. To construct the matrix, a row-by-row storage algorithm is used with searching for elements through a hash table, which reduces the amount of memory used, the number of thread synchronizations and can significantly speed up calculations. The algorithm in question has been successfully applied in the Beren3D code.

In this paper, a linear second-order finite difference scheme is proposed for the Allen-Cahn equation with a general positive mobility. The Crank-Nicolson scheme and Taylor's formula are used for temporal discretization, and the central finite difference method is used for spatial approximation. The discrete maximum bound principle (MBP), the discrete energy stability and L^∞-norm error estimation are discussed, respectively. Finally, some numerical examples are presented to verify our theoretical results.

We present in this work a convergence analysis of a Finite Difference method for solving on quadrilateral meshes 2D-flow problems in homogeneous porous media with a full permeability tensor. We start with the derivation of the discrete problem by using our finite difference formula for a mixed derivative of second order. A result of existence and uniqueness of the solution for that problem is given via the positive definiteness of its associated matrix. Their theoretical properties, namely, stability on the one hand (with the associated discrete energy norm) and error estimates (with L²-norm, relative L²-norm and L^∞-norm ) are investigated. Numerical simulations are shown.

The manuscript presents the results of an application of a numerical method to solve one-dimensional hyperbolic equations. These equations simulate the dynamics of a liquid in a pipe with varying cross-sections. The equations are written in terms of pressure-head and discharge. Radial-basis functions and least-squares optimization are used for the numerical simulation. This numerical method is specialized for working with arbitrary nodal distribution in the problem domain. The basics of the application of the numerical method were introduced in our previous work. In the current work, we updated previously applied methods by means of getting rid of the time-marching approach and applying another adaptive refinement technique. Three cases of the simulations of the reservoir-pipe-valve system are described, indicating that the sharp time-gradient phenomenon is reproduced by the model.

The paper presents a Symmetric Hyperbolic Thermodynamically Compatible model of a saturated porous medium for the case of finite deformations and its linearisation for the description of small amplitude seismic wave fields in porous media saturated with fluid. The model allows us to describe wave processes for different phase states of the saturating fluid during its transition from solid to liquid state, for example during thawing of permafrost and decomposition of gas hydrates under the influence of temperature. To numerically solve the governing equations of the model, a finite difference method on staggered grids has been developed. It was used to perform test calculations for a model of the medium containing a layer of gas hydrate in a homogeneous elastic medium. The study showed that the characteristics of the wave fields in saturated porous media depend significantly on the porosity, which varies with temperature.

This work provides a high accuracy analysis of the reduced Adini Stokes element method developed in [7] for the Brinkman model. We show that this method is uniformly convergent for the velocity with convergence order O(h ²) in a mesh- and parameter-dependent norm over general quasi-uniform rectangular meshes. A proper postprocessing technique is also proposed to improve the precision of the pressure. Numerical examples confirm our theory.

The article investigates the application of hyperbolization for parabolic equations to the material and thermal balances' equations for a mathematical model of oxidative regeneration of a spherical catalyst grain with detailed kinetics. The initial spherical grain model is constructed using a diffusion approach. It is a nonlinear system of differential equations in a spherical coordinate system. The material balance of the gas phase is described by diffusion-convection-reaction equations with source terms compiled for concentrations of substances of the gas phase; the balance of the solid phase is represented by nonlinear ordinary differential equations. The thermal balance equation of the catalyst grain is the thermal conductivity equation with an inhomogeneous term corresponding to the grain heating during a chemical reaction. Slow processes of heat and mass transfer in combination with fast chemical reactions lead to significant difficulties in the development of a computational algorithm. Hyperbolization of the parabolic equations is applied to avoid the computational complication. It consists in the introduction of a second time derivative multiplied by a small parameter, in order to expand the stability area of the computational algorithm. An explicit three-layer difference scheme is constructed for the modified model. It is implemented in the form of a software module. The convergence analysis of the developed algorithm is presented. A comparative analysis of the new computational algorithm with the previously constructed one is carried out. The advantage of the new algorithm while maintaining the order of accuracy is shown. The result of the implemented new algorithm is the profiles of the distribution of temperature and substances along the radius of the catalyst grain.