Local Discontinuous Galerkin Methods for One-Dimensional Second Order Fully Nonlinear Elliptic and Parabolic Equations

This paper is concerned with developing accurate and efficient nonstandard discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$ , two independent functions $q^-$ and $q^+$ are introduced to approximate one-sided derivatives of $u$ . Similarly, to capture the discontinuities of the second order derivative $u_{xx}$ , four independent functions $p^{- -}, p^{- +}, p^{+ -}$ , and $p^{+ +}$ are used to approximate one-sided derivatives of $q^-$ and $q^+$ . The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$ . An easy to verify set of criteria for constructing “good” numerical operators is also proposed. It consists of consistency and generalized monotonicity. To ensure such a generalized monotonicity property, the crux of the construction is to introduce the numerical moment in the numerical operator, which plays a critical role in the proposed LDG framework. The generalized monotonicity gives the LDG methods the ability to select the viscosity solution among all possible solutions. The proposed framework extends a companion finite difference framework developed by Feng and Lewis (J Comp Appl Math 254:81–98, 2013) and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. Numerical experiments are also presented to demonstrate the accuracy, efficiency and utility of the proposed LDG methods.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

Discrete Fundamental Solution Preconditioning for Hyperbolic Systems of PDE

We present a new preconditioner for the iterative solution of systems of equations arising from discretizations of systems of first order partial differential equations (PDEs) on structured grids. Such systems occur in many important applications, including compressible fluid flow and electromagnetic wave propagation. The preconditioner is a truncated convolution operator, with a kernel that is a fundamental solution of a difference operator closely related to the original discretization. Analysis of a relevant scalar model problem in two spatial dimensions shows that grid independent convergence is obtained using a simple one-stage iterative method. As an example of a more involved problem, we consider the steady state solution of the non-linear Euler equations in a two-dimensional, non-axisymmetric duct. We present results from numerical experiments, verifying that the preconditioning technique again achieves grid independent convergence, both for an upwind discretization and for a centered second order discretization with fourth order artificial viscosity.     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.     

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.     

求解非线性期权定价模型的自适应小波同伦摄动技术

Leland 模型是在考虑交易费用的情况下,对 Black - Scholes 模型进行修改得到的非线性期权定价模型. 本文针对 Leland 模型,提出了一种求解非线性动力学模型的自适应多尺度小波同伦摄动法. 该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性期权定价模型方程自适应离散为非线性常微分方程组; 然后将用于求解非线性常微分方程组的同伦摄动技术和小波变换的动态过程相结合,构造了求解 Leland 模型的自适应数值求解方法. 数值模拟结果验证了该方法在数值精度和计算效率方面的优越性.     

Solving differential equations with Fourier series and Evolution Strategies

A novel mesh-free approach for solving differential equations based on Evolution Strategies (ESs) is presented. Any structure is assumed in the equations making the process general and suitable for linear and nonlinear ordinary and partial differential equations (ODEs and PDEs), as well as systems of ordinary differential equations (SODEs). Candidate solutions are expressed as partial sums of Fourier series. Taking advantage of the decreasing absolute value of the harmonic coefficients with the harmonic order, several ES steps are performed. Harmonic coefficients are taken into account one by one starting with the lower order ones. Experimental results are reported on several problems extracted from the literature to illustrate the potential of the proposed approach. Two cases (an initial value problem and a boundary condition problem) have been solved using numerical methods and a quantitative comparative is performed. In terms of accuracy and storing requirements the proposed approach outperforms the numerical algorithm.     

A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L ² stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.     

Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation

In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme.     

Numerical Analysis of Second Order,Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank–Nicolson method for time discretization, projection method for Navier–Stokes equations, as well as several implicit–explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.     

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.     

On two upwind finite-difference schemes for hyperbolic equations in non-conservative form

Several finite difference-schemes for approximating solutions of initial value problems associated with systems of linear hyperbolic differential equations are considered. Common features of the schemes is the approximation of the space-like derivatives according to the behavior of the characteristics (upwind schemes for hyperbolic equations). The analysis of standard properties (consistency, stability, convergence, dissipativity, phase error) of finite difference schemes is performed. In addition, extensions of certain upwind schemes to nonlinear equations, extensions to several space-like dimensions by splitting methods and two implicit finite difference schemes are considered.     

Mean field games of energy storage devices: A finite difference analysis

This paper considers a mean field game of energy storage devices (ESDs) in power systems, where electrovalency is affected by the storage population. The competition processes of ESDs are characterized by a pair of coupled partial differential equations (PDEs). The so-called mean field equilibrium (MFE) is obtained by resolving the PDEs. The resulting MFE strategy is then broadcast to all local controllers in order to independently determine a control input for each device. Since PDEs are difficult to be explicitly calculated, a pair of difference schemes are designed for approximating the exact solutions of the PDEs. Then the sufficient condition of stability is analyzed. Based on the proposed difference schemes, a difference iterative scheme is designed for numerically calculating the MFE. Some simulation results show the effectiveness of the proposed iterative schemes.     

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L ² stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.     

Numerical solution of PDEs via integrated radial basis function networks with adaptive training algorithm

This paper develops a mesh-free numerical method for solving PDEs, based on integrated radial basis function networks (IRBFNs) with adaptive residual subsampling training scheme. The multiquadratic function is chosen as the transfer function of the neurons. The nonlinear algebraic equation systems for weights training are solved by Levenberg–Marquardt algorithm. The performance of the proposed method is demonstrated in numerical examples by approximating several functions and solving nonlinear PDEs. The result of numerical experiments shows that the IRBFNs with the adaptive procedure requires less neurons to attain the desired accuracy than conventional radial basis function networks.     

Fast multilevel augmentation methods for nonlinear boundary value problems

We develop a multilevel augmentation method for solving nonlinear boundary value problems. We first describe the multilevel augmentation method for solving nonlinear operator equations of the second kind which has been proposed in our recent paper Chen et al. (2011) [16], and then apply it to solving the nonlinear two-point boundary value problems of second-order differential equations. The theoretical analysis of convergence order and computational complexity are proposed. Finally numerical experiments are presented to confirm the theoretical estimates and illustrate the efficiency of the method.     

Toward designing intelligent PDEs for computer vision: An optimal control approach

Many computer vision and image processing problems can be posed as solving partial differential equations (PDEs). However, designing a PDE system usually requires high mathematical skills and good insight into the problems. In this paper, we consider designing PDEs for various problems arising in computer vision and image processing in a lazy manner: learning PDEs from training data via an optimal control approach. We first propose a general intelligent PDE system which holds the basic translational and rotational invariance rule for most vision problems. By introducing a PDE-constrained optimal control framework, it is possible to use the training data resulting from multiple ways (ground truth, results from other methods, and manual results from humans) to learn PDEs for different computer vision tasks. The proposed optimal control based training framework aims at learning a PDE-based regressor to approximate the unknown (and usually nonlinear) mapping of different vision tasks. The experimental results show that the learnt PDEs can solve different vision problems reasonably well. In particular, we can obtain PDEs not only for problems that traditional PDEs work well but also for problems that PDE-based methods have never been tried before, due to the difficulty in describing those problems in a mathematical way.     

One-shot methods in function space for PDE-constrained optimal control problems

We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.     

On a System of Adaptive Coupled PDEs for Image Restoration

In this paper, we consider a coupled system of partial differential equations (PDEs) based model for image restoration. Both the image and the edge variables are incorporated by coupling them into two different PDEs. It is shown that the initial-boundary value problem has global in time dissipative solutions (in a sense going back to P.-L. Lions), and several properties of these solutions are established. Some numerical examples are given to highlight the denoising nature of the proposed model along with some comparison results.