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17 pages, 329 KiB

Open AccessArticle

Correct and Stable Algorithm for Numerical Solving Nonlocal Heat Conduction Problems with Not Strongly Regular Boundary Conditions

by Makhmud A. Sadybekov and Irina N. Pankratova

Mathematics 2022, 10(20), 3780; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203780 - 13 Oct 2022

Cited by 1 | Viewed by 756

Abstract

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. [...] Read more.

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid

L_{2}^{h}

-norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

14 pages, 1488 KiB

Open AccessArticle

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

by Jorge E. Macías-Díaz, Nuria Reguera and Adán J. Serna-Reyes

Mathematics 2021, 9(21), 2727; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212727 - 27 Oct 2021

Cited by 4 | Viewed by 1267

Abstract

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The [...] Read more.

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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20 pages, 322 KiB

Open AccessArticle

Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

by Maria Jesus Moreta

Mathematics 2021, 9(18), 2225; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182225 - 10 Sep 2021

Viewed by 1800

Abstract

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization [...] Read more.

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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17 pages, 355 KiB

Open AccessArticle

Efficient Time Integration of Nonlinear Partial Differential Equations by Means of Rosenbrock Methods

by Isaías Alonso-Mallo and Begoña Cano

Mathematics 2021, 9(16), 1970; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161970 - 17 Aug 2021

Cited by 3 | Viewed by 1468

Abstract

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use [...] Read more.

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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40 pages, 3307 KiB

Open AccessArticle

Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

by Irene Gómez-Bueno, Manuel Jesús Castro Díaz, Carlos Parés and Giovanni Russo

Mathematics 2021, 9(15), 1799; https://0-doi-org.brum.beds.ac.uk/10.3390/math9151799 - 29 Jul 2021

Cited by 9 | Viewed by 1710

Abstract

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify [...] Read more.

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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22 pages, 1538 KiB

Open AccessArticle

A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

by Adán J. Serna-Reyes, Jorge E. Macías-Díaz and Nuria Reguera

Mathematics 2021, 9(12), 1412; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121412 - 18 Jun 2021

Cited by 3 | Viewed by 1746

Abstract

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz [...] Read more.

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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24 pages, 433 KiB

Open AccessArticle

Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

by Isaías Alonso-Mallo and Ana M. Portillo

Mathematics 2021, 9(10), 1113; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101113 - 14 May 2021

Cited by 1 | Viewed by 1601

Abstract

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a [...] Read more.

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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20 pages, 796 KiB

Open AccessArticle

Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?

by Begoña Cano and Nuria Reguera

Mathematics 2021, 9(9), 1008; https://0-doi-org.brum.beds.ac.uk/10.3390/math9091008 - 29 Apr 2021

Cited by 1 | Viewed by 1431

Abstract

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order [...] Read more.

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost. Full article

(This article belongs to the Special Issue Numerical Methods for Evolutionary Problems)

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