Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Since hyperbolic equations usually propagate discontinuities of initial or boundary conditions into the domain on which the solution lives or can develop discontinuities even in the presence of smooth data, problems of this type constitute a severe challenge for both theory and numerics of PDE constrained optimization.     

Discontinuous Galerkin solution of compressible flow in time-dependent domains

This work is concerned with the simulation of inviscid compressible flow in time-dependent domains. We present an arbitrary Lagrangian–Eulerian (ALE) formulation of the Euler equations describing compressible flow, discretize them in space by the discontinous Galerkin method and introduce a semi-implicit linearized time stepping for the numerical solution of the complete problem. Special attention is paid to the treatment of boundary conditions and the limiting procedure avoiding the Gibbs phenomenon in the vicinity of discontinuities. The presented computational results show the applicability of the developed method.     

A semi-analytical iterative technique for solving nonlinear problems

We present a semi-analytical iterative method for solving nonlinear differential equations. To demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial/boundary conditions. In each of the examples we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a high order of convergence.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.     

Artificial neural networks for solving ordinary and partialdifferential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE), to systems of coupled ODE and also to partial differential equations (PDE). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galerkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.     

Calculation of the three-dimensional boundary layer with solution of all three momentum equations

The paper describes the application of a calculation procedure to three flow situations which can be characterised as three-dimensional boundary layers. Unlike most of the published methods, the present procedure solves all momentum equations and takes full account of the pressure variation in directions normal to the main-flow direction. The applications demonstrate that, when the boundary conditions exhibit certain discontinuities, only the solution of all three momentum equations can give satisfactory accuracy. The results of the present calculations are compared with available similarity solutions wherever possible.     

Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

Upwinding in the method of lines

The method of lines (MOL) is a procedure for the numerical integration of partial differential equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences (FDs). If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations (ODEs) results through the algebraic approximation of the spatial derivatives.If the PDEs are strongly convective (strongly hyperbolic), they can propagate sharp fronts and even discontinuities, which are difficult to resolve in space. Experience has demonstrated that for these systems, some form of upwinding is generally required when replacing the spatial derivatives with algebraic approximations. Here we investigate the performance of various forms of upwinding to provide some guidance in the selection of upwind methods in the MOL solution of strongly convective PDEs.     

An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations

We present a double Legendre spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomo-geneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. One numerical application of how to use these methods is described. Numerical results obtained compare favorably with those of the analytical solution. Accurate double Legendre spectral approximations for Poisson' and Helmholtz' equations are also noted.     

On initial value and terminal value problems for Hamilton–Jacobi equation

First order partial differential equations (PDE) are often the main tool to model problems in optimal control, differential games, image processing, physics, etc. Dependent upon the particular application, the boundary conditions are specified either at the initial time instant, leading to an initial value problem (IVP), or at the terminal time instant, leading to a terminal value problem (TVP). The IVP and TVP have in general different solutions. Thus introducing a new model in terms of a first order PDE one has to consider both possibilities of IVP and TVP, unless there is a direct physical indication. In this paper we also particularly answer the following question: how should the initial value at the initial surface and the terminal value at the terminal surface be coordinated in order to generate the same solution? One may expect that for a given initial value the consistent terminal value is the value of the IVP solution at the terminal surface. The second (time-varying) example in this paper shows that, generally, this is not true for non-smooth initial conditions. We discuss also the difference between the IVP and TVP formulations, the connection between the Hamiltonians arising in IVP and TVP.     

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.     

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.     

Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.     

Die Lösung des Anfangs-Randwertproblems für die Wärmeleitungsgleichung im ℝ3 mit einer Integralgleichungs-methode nach dem Rotheverfahrenmit einer Integralgleichungs-methode nach dem Rotheverfahren

We are looking for a solution of the initial boundary value problem for the threedimensional heat equation in a compact domain with a boundary of continous curvature. We use Rothe's line method, which works by discretisation of the time variable. For every time step there remains an elliptic boundary value problem, which is solved by means of an integral equation. The so obtained approximate solutions converge to the exact solution of the original problem. In case of a sphere we find a simple error estimate for the approximation. For two initial conditions the practical computations show, that the integral equations method yields useful results with relative small effort.     

Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.     

Weak discontinuity annihilation in solidification modelling

Discontinuous behaviour provides substantial obstacles to the efficient application of mesh based numerical techniques. Accounting for strong discontinuities is presently of particular interest to the finite element research community with for example the development of cohesive and enriched elements to cater for material separation. Although strong discontinuities are of importance, of equal if not of greater interest and the focus in this paper, are weak discontinuities, which are present at any material change. A recent innovation for accounting for weak discontinuities has been the discovery of non-physical variables which are founded and defined using transport equations.This paper is concerned with the application of the non-physical approach to solidification modelling in the presence of more than one material discontinuity. A typical feature of the enthalpy-temperature response in solidification is discontinuities at phase transition temperatures as a consequence of phase change and latent heat release. In these circumstances, depending on the conditions that prevail, an element in a finite element mesh can have more than one discontinuity present.Presented in the paper is a methodology that can cater for multiple discontinuities. The non-physical approach permits the precise removal of weak discontinuities arising in the governing transport equations. In order to facilitate the application of the approach the finite element equations are presented in the form of weighted transport equations. The method utilises a non-physical form of enthalpy that possesses a remarkable source distribution like property at a discontinuity. It is demonstrated in the paper that it is through this property that multiple discontinuities can be exactly removed from an element so facilitating the use of continuous approximations.The new methodology is applied to a range of simple problems to provide an in-depth treatment and for ease of understanding to demonstrate the methods remarkable accuracy and stability.     

Classical necessary optimality conditions in discrete optimal control problems with nonlocal conditions

We consider an optimal control problem in which the system’s state is described by a system of difference equations with nonlocal (two-point) conditions; this problem includes, as particular cases, the initial value problem (Cauchy problem) and different types of boundary value problems. It is assumed that the admissible controls take values from an open set. The first and second functional variations are calculated; these variations are used to express first and second order necessary optimality conditions in the classical sense for discrete optimal control problems.     

High Order Finite Difference and Finite Volume Methods for Advection on the Sphere

Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significant potential gain in computational efficiency. We here present two classes of high order methods for discretization on the surface of a sphere, first finite difference schemes satisfying the summation-by-parts property on the cube sphere grid, secondly finite volume discretizations on unstructured grids with polygonal cells. Furthermore, we also implement the seventh order accurate weighted essentially non-oscillatory (WENO7) scheme for the cube sphere grid. For the finite difference approximation, we prove a stability estimate, derived from projection boundary conditions. For the finite volume method, we develop the implementational details by working in a local coordinate system at each cell. We apply the schemes to compute advection on a sphere, which is a well established test problem. We compare the performance of the methods with respect to accuracy, computational efficiency, and ability to capture discontinuities.     

Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis

We consider a class of Boussinesq systems of Bona-Smith type in two space dimensions approximating surface wave flows modelled by the three-dimensional Euler equations. We show that various initial-boundary-value problems for these systems, posed on a bounded plane domain are well posed locally in time. In the case of reflective boundary conditions, the systems are discretized by a modified Galerkin method which is proved to converge in L ² at an optimal rate. Numerical experiments are presented with the aim of simulating two-dimensional surface waves in realistic (plane) domains with a variety of initial and boundary conditions, and comparing numerical solutions of Bona-Smith systems with analogous solutions of the BBM-BBM system.