Dimensional Parabolic Partial Differential Equations Research Articles

On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in Zang et al. (2020) [17] offered a flexible and computationally efficient approach to tackle this problem defined on arbitrary domains by leveraging the weak solution. WAN reformulates the PDE problem as a generative adversarial network, where the weak solution (primal network) and the test function (adversarial network) are parameterized by the multi-layer deep neural networks (DNNs). However, it is not yet clear whether DNNs are the most effective model for the parabolic PDE solutions as they do not take into account the fundamentally different roles played by time and spatial variables in the solution. To reinforce the difference, we design a novel so-called XNODE model for the primal network, which is built on the neural ODE (NODE) model with additional spatial dependency to incorporate the a priori information of the PDEs and serve as a universal and effective approximation to the solution. The proposed hybrid method (XNODE-WAN), by integrating the XNODE model within the WAN framework, leads to significant improvement in the performance and efficiency of training. Numerical results show that our method can reduce the training time to a fraction of that of the WAN model.

Towards Fast Weak Adversarial Training to Solve High Dimensional Parabolic Partial Differential Equations Using Xnode-Wan

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in (Y.Zang et al., 2020) offered a flexible and computationally efficient approach to tackle this problem defined on arbitrary domains by leveraging the weak solution. WAN reformulates the PDE problem as a generative adversarial network, where the weak solution (primal network) and the test function (adversarial network) are parameterized by the multi-layer deep neural networks (DNNs). However, it is not yet clear whether DNNs are the most effective model for the parabolic PDE solutions as they do not take into account the fundamentally different roles played by time and spatial variables in the solution. To reinforce the difference, we design a novel so-called XNODE model for the primal network, which is built on the neural ODE (NODE) model with additional spatial dependency to incorporate the a priori information of the PDEs and serve as a universal and effective approximation to the solution. The proposed hybrid method (XNODE-WAN), by integrating the XNODE model within the WAN framework, leads to significant improvement in the performance and efficiency of training. Numerical results show that our method can reduce the training time to a fraction of that of the WAN model.

Equilibrium strategies in a multiregional transboundary pollution differential game with spatially distributed controls

We analyse a differential game with spatially distributed controls to study a multiregional transboundary pollution problem. The dynamics of the state variable (pollution stock) is defined by a two dimensional parabolic partial differential equation. The control variables (emissions) are spatially distributed variables. The model allows for a, possibly large, number of agents with predetermined geographical relationships. For a special functional form previously used in the literature of transboundary pollution dynamic games we analytically characterize the feedback Nash equilibrium. We show that at the equilibrium both the level and the location of emissions of each region depend on the particular geographical relationship among agents. We prove that, even in a simplified model, the geographical considerations can modify the players’ optimal strategies and therefore, the spatial aspects of the model should not be overlooked.

Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation

Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. Finite difference method is a popular numerical method which has been applied extensively to solve partial differential equations. A well known type of this method is the Classical Crank-Nicolson scheme which has been used by different researchers. In this work, we present a modified Crank-Nicolson scheme resulting from the modification of the classical Crank-Nicolson scheme to solve one dimensional parabolic equation. We apply both the Classical Crank-Nicolson scheme and the modified Crank-Nicolson scheme to solve one dimensional parabolic partial differential equation and investigate the results of the different schemes. The computation and results of the two schemes converges faster to the exact solution. It is shown that the modified Crank-Nicolson method is more efficient, reliable and better for solving Parabolic Partial differential equations since it requires less computational work. The method is stable and the convergence is fast when the results of the numerical examples where compared with the results from other existing classical scheme, we found that our method have better accuracy than those methods. Some numerical examples were considered to verify our results.

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by $O( d \, \epsilon^{-(4+\delta)})$ for any $\delta\in(0,\infty)$, where $d$ is the dimensionality of the problem and $\epsilon\in(0,\infty)$ is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of $100$-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature.

A Hybrid Solutions Method for Solving One Dimensional Parabolic Partial Differential Equations

A Hybrid Solutions Method for Solving One Dimensional Parabolic Partial Differential Equations

The analytical solution of 2D electromagnetic wave equation for eddy currents in the cylindrical solid rotor structures

The paper presents the closed-form solution of two dimensional (2D) electromagnetic wave equation for eddy current problems in cylindrical structures. The magnetic field calculation is a complex issue for electrical machines especially. The paper provides an analytical solution for solid cylindrical structures with linear magnetization. This approach is extended to the electrical machines with the solid rotors. Multilayered cylindrical geometry is used for the solution. Two dimensional parabolic partial differential equation is solved for each layer. According to the analytical solution, eddy current and eddy current losses are calculated in the solid region and compared to the results of the finite element model.

Polynomials Approximation Method for Solving Parabolic Partial Differential Equations

A new numerical method based on the approximation of polynomials is here proposed for solving the one dimensional parabolic partial differential equation arising from unsteady state flow of heat subject to initial and boundary conditions. The method results from discretization of the parabolic partial differential equation which leads to the production of a system of algebraic equations. By solving the system of algebraic equations we obtain the problem approximate solutions.

A computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes

Abstract In this article, trigonometric B-spline collocation method is used to compute the numerical solution of nonlinear Fisher’s equation, in which the nonlinear term is locally linearized. This equation arises in many biological and chemical processes. The consistency of the proposed method is shown using the concept of stability and convergence and to validate the numerical scheme, the obtained theoretical results for convergence are given. We have simulated certain numerical examples of the Fisher’s equation and compared their results with the exact solutions of the problems. The numerical results are found to be in good agreement with the exact solution. The accuracy and efficiency of the method are discussed by computing L 2 and L ∞ norm which are represented in the forms of tables and figures.

Parallel Performance Comparison of Alternating Group Explicit Method Between Parallel Virtual Machine and Matlab Distributed Computing for Solving Large Sparse Partial Differential Equations

This paper presents the parallel implementation and performance analysis of a large sparse matrix of one dimensional parabolic Partial Differential Equation (PDE) between Parallel Virtual Machine (PVM) and Matlab Distributed Computing (MDC) platform. PVM is a software to perform parallel programming using heterogeneous or homogeneous clusters of parallel computing system connected by a logical network system to users as virtual memory parallel computer system. MDC is a scientific library programming language. Alternating Group Explicit (AGE) method is used to solve the parabolic equation with large sparse matrix. The distributed parallel algorithm of AGE method is implemented on the PVM with RedHat Linux operation and MDC Version 2011a. The parallel performance evaluation (PPE) is compared between the two platforms in terms of time execution, speedup, efficiency, effectiveness, and temporal performance. From the performance analysis, we can conclude that PVM is better compared to MDC.

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work [1–3] from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.

Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions

A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank–Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion–reaction equation is used for testing with the Dirichlet and mixed Dirichlet–Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

AbstractTraditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Alternating direction finite volume element methods for 2D parabolic partial differential equations

AbstractOn the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L2 norm or H1 semi‐norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Accelerating technique for implicit finite difference schemes for two dimensional parabolic partial differential equations

In this paper we define a new accurate fast implicit method for the finite difference solution of the two dimensional parabolic partial differential equations with first level condition, which may be obtained by any other method. The stability region is discussed. The suggested method is considered as an accelerating technique for the implicit finite difference scheme, which is used to find the first level condition. The obtained results are compared with some famous finite difference schemes and it is in satisfactory agreement with the exact solution.

Global Extrapolation of a First Order Splitting Method

This note deals with the numerical solution of multi-space dimensional parabolic partial differential equations and advocates classic global Richardson extrapolation for splitting methods. The pape...

Contractivity of locally one-dimensional splitting methods

The aim of this paper is to study contractivity properties of two locally one-dimensional splitting methods for non-linear, multi-space dimensional parabolic partial differential equations. The term contractivity means that perturbations shall not propagate in the course of the time integration process. By relating the locally one-dimensional methods with contractive integration formulas for ordinary differential systems it can be shown that the splitting methods define contractive numerical solutions for a large class of non-linear parabolic problems without restrictions on the size of the time step.

Block methods for parabolic equations

Block methods for the finite difference solution of linear one dimensional parabolic partial differential equations are considered. These schemes use two linear multistep formulae which, when applied simultaneously, advance the numerical solution by two time steps. No special starting procedure is required for their implementation. By careful choice of the coefficients in these formulae, all of the block methods derived in this paper are unconditionally stable and have high order accuracy. In addition, some of these schemes are suitable for problems involving a discontinuity between the initial and boundary conditions. The results of numerical experiments on two test problems are presented.

Collocation-$H^{ - 1} $-Galerkin Method for Parabolic Problems with Time Dependent Coefficients

A collocation-$H^{ - 1} $-Galerkin procedure is introduced and analyzed for a one space dimensional parabolic partial differential equation. Special attention is devoted to the behavior of the error due to the time dependent nature of the coefficients. Global error estimates are derived.