Discontinuous Source Term Research Articles

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

A numerical solution of singularly perturbed Fredholm integro-differential equation with discontinuous source term

In this paper, we investigate a singularly perturbed Fredholm integro-differential problem with a discontinuous source term, leading to the formation of interior layers in the solution at the point of discontinuity. We apply the exponentially fitted mesh method to solve the problem. Our analysis demonstrates that the method exhibits almost first-order convergence in the maximum norm, independent of the diffusion parameter. Numerical experiments are conducted to verify the validity of our theoretical results and they support the accuracy of our estimates.

Numerical Treatment of a Two-Parameter Singularly Perturbed Elliptic Problem with Discontinuous Convection and Source Terms

AbstractIn this paper, we address a two-parameter singularly perturbed convection-reaction-diffusion 2-D problem. We also consider that the convection and source terms are discontinuous in space. Due to these discontinuities and the presence of perturbation parameters, solutions to such problems show boundary and interior layers. In this study, we have carried out a numerical approach using a finite-difference technique with an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the best performance of the proposed method and its agreement with the theoretical analysis.

Hybrid Difference Scheme for Singularly Perturbed Differential Equation with Discontinuous Source Term

Hybrid Difference Scheme for Singularly Perturbed Differential Equation with Discontinuous Source Term

A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term

A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term

An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source terms

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings.

An equidistributed grid-based second-order scheme for a singularly perturbed Fredholm integro-differential equation with an interior layer

In this study, we introduce and evaluate the effectiveness of the central finite difference coupled with a composite trapezoidal rule on adapted meshes in solving a second-order singularly perturbed Fredholm integro-differential equation with a discontinuous source term. A grid equidistribution technique was employed to address the challenge posed by the perturbation parameter in classical numerical methods. Our approach demonstrated superior precision compared to existing schemes in the literature and achieved second-order uniform convergence.

Potentiality of the HOFiD_bvp code in solving different kind of second order boundary value problems

The aim of this paper is to present the potentiality of the code HOFiD_bvp based on high order finite difference, following the approach of the boundary value methods and the upwind methods. Strategies, as variable stepsize and variable order, that helps to handle with solutions of singular perturbation problems, so as the continuation strategy applied for solving nonlinear problems, will be discussed. Now, the method is suitable to solve second order scalar boundary value problems, not only singularly perturbed with one and two parameters, but also problems with discontinuous source terms. Concerning that point some numerical tests will be taken in consideration in order to show the potentiality of the code.

A Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term

A Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term

Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomenon called detonation will occur provided that the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure and from the characteristic boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary, and exothermic reaction and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains.

A fitted numerical method for a singularly perturbed Fredholm integro-differential equation with discontinuous source term

In this article, a singularly perturbed second-order Fredholm integro-differential equation with a discontinuous source term is examined. An exponentially-fitted numerical method on a Shishkin mesh is applied to solve the problem. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Some numerical results are given, which validate the theoretical results.

NIPG Method on Shishkin Mesh for Singularly Perturbed Convection–Diffusion Problem with Discontinuous Source Term

In this paper, we provide numerical approaches for singularly perturbed convection–diffusion problem with a discontinuous source term. As a result of the discontinuity in the source term, the problem highlights an exponential-type boundary layer and a weak interior layer. To handle both the layers at the same time, we design a nonsymmetric discontinuous Galerkin finite element technique with interior penalties (NIPG). The problem is solved using the standard Shishkin mesh. In the DG norm, uniform error estimates are derived for the problem. To verify the theoretical estimates, numerical tests are carried out.

Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space

Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.