Fourth‐order finite difference method for 2D parabolic partial differential equations with nonlinear first‐derivative terms

Domain decomposition and expanded mixed method for parabolic partial differential equations

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

In this paper, we verify the convergence and stability of implicit (modified) finite difference scheme. Knowing fully that consistency and stability are very important criteria for convergence, we have prove the stability of the modied implicit scheme using the von Newmann method and also verify the convergence by comparing the numerical solution with the exact solution. The results shows that the schemes converges even as the step size is rened.

Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

: This document considers two adaptive finite element techniques for parabolic partial differential equations (PDEs) that are based on using error estimates to control mesh refinement. One technique is a method of lines approach that uses a Galerkin method to discretize the PDEs in space and implicit multi-step integration in time. Spatial elements are added and deleted in regions of high and low error and are all advanced with the same sequence of varying time steps. The second technique is a local refinement method that uses Galerkin approximations in both space and time. Fine grids of space-time elements are added to coarser grids and the problem is recursively solved in regions of high error. (Author)

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

In Part I of this paper (SIAM J. Numer. Anal., 15 (1978), pp. 1212–1224), novel algorithms were introduced for solving parabolic differential equations in which high-frequency components occurred in the solution and for which $A_0 $-stable methods, exemplified by the classical Crank-Nicolson method, were less than satisfactory. The algorithms presented were based on a simple extrapolation of the Backward Euler method which produced $L_0 $ -stability. In all cases the algorithms presented were second order accurate in time. In the present paper some of the previous algorithms are generalized to higher orders of accuracy. In particular, a more detailed consideration of the second order algorithm is given which leads naturally to methods of orders three and four. The novel algorithms are tested on a heat equation with constant coefficients in which a discontinuity between the initial values and boundary values exists.

For Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and a posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Iterative Methods for Nonlinear Equations.- Discretization Methods for Convection-Dominated Problems.- Appendices.

We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) as well as choosing the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time amd initial condition is introduced and shown to be stable. Finally, a Reduced Basis Method (RBM) is introduced that is able to use parameter functions also for the initial condition. Corresponding numerical results are shown.

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

The parabolic equation (PE) method is widely used in radiowave propagation predictions. It has the advantages of high efficiency and stability, but it will lead to greater predicting errors in some situations, because the effects of transverse terrain gradients are not modeled. This problem can be solved by extending the 2D PE to the three-dimensional (3D) PE. However, the computing efficiency will degrade because of large scale matrix operations. In this paper, a new method is presented, in which the 3D PE is decomposed into two 2D PEs. It increases the computational efficiency and accuracy effectively. To verify the capability of the proposed method in radiowave propagation prediction, an experiment platform was set up. The computational results using this new method are compared with the experimental and Method of Moment (MoM) numerical computational results. Good agreements are achieved in the comparison. The study of radiowave propagation under complex environments is vitally important for improving the effective signal coverage in wireless communication and television broadcasting with plenty of urban buildings as well as mountainous terrains. It presents a big challenge due to the multiple wave propagation mechanisms involving the line of sight, diffraction and scattering propagation and the characteristic of large scale computation. The major issues refer to how to keep the computational accuracy and conducting the large scale computation at the same time. To meet the request for large scale computation of actual engineering problems, the modified empirical formulas are often used for analysis and computation with the drawback of poor computational accuracy, and the variance of the predicted propagation loss is generally 2 to 5 dB for the flat terrain, but for the complex terrain, the variance of the predicted propagation loss is usually about 5 to 10 dB. Many computational methods have been developed to improve the predicted accuracy. The commonly used methods are geometric diffraction methods, path integration methods and FDTD. It is hard to meet the requirement of engineering computation because these methods are computationally expensive and require high data accuracy of the terrain database at the same time (1). The Parabolic Equation (PE) method is a computational method for radiowave propagation loss besides the full wave method and ray tracing method, and has become a research focus. The PE method has high potential in radiowave propagation of mobile communication and broadcasting and television signal coverage as it can not only deal with complex terrain boundary conditions accurately, but also deal with complex atmosphere waveguide structure. Compared with the full wave method, the PE method deals with an unidirectional wave propagation, in which the equation can be solved by the step marching method in the propagation direction. The PE method can reduce the computational complexity of the full wave equation.

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.