Novel numerical methods for second order partial Volterra integro-differential equations in two dimensional space

In this article we present the first and second order numerical schemes for the solution of initial value problems of the Boltzmann-Peierls equation (BPE). We also modify the numerical schemes for the solution of initial and boundary value problems (IBVP) of its derived hyperbolic moment system. BPE is an integro-differential equation which describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE, while it still retain all the properties of the original BPE. Using special coordinates, we get a further reduction of the kinetic equation in one space dimension. We introduce the discrete-velocity model of the reduce BPE in one space dimension. This discrete-velocity model can be discretized in space and time by using finite volume schemes. We derive both first and second order explicit upwind and central schemes for the discrete-velocity kinetic equation as well as for the derived moment system. We use the kinetic approach in order to prescribe boundary conditions for the IBVP of the moment system. Several numerical test cases are considered in order to validate the theory.

This paper presents an elementary introduction to linear Volterra integral and integro-differential equations. It is demonstrated that the theory of existence, uniqueness, dimensionality of the solution space, and the variation of parameters formula are virtually indistinguishable from the corresponding elementary theory of ordinary differential equations. A review of some current problems is also included.

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

In this paper, a hybrid technique based on the combining of differential transform approach, the Laplace transformation, and Pade approximant is successfully implemented for solving Volterra Integro differential (VID) equations subject to appropriate conditions. The proposed method known as modified differential transform method (MDTM) enlarges the domain of convergence with less computational time. One of the interesting features of this algorithm is the ability to produce an analytic approximate solution in convergent power series form with few numbers of computable terms. Preliminary results on some test problems show that the new algorithm is efficient and promising for Volterra integro-differential equations and can be applied as alternatives for the integral and integro-differential (IID) equation that have no analytical solution.

In the last two decades the theory of Volterra integral equations and of integro-differential equations has developed extensively. New classes of methods for the numerical solution of such equations have been developed and at the same time there have been advances in the qualitative theory of these equations. More frequent use is being made of Volterra equations to model various physical and biological phenomenon as new insight has occurred into the asymptotic behaviour of solutions. In consequence, there has emerged a need for reliable and efficient methods for the numerical treatment of such equations. This thesis is concerned with an aspect of numerical solution of Volterra integral and integro-differential equations. In Chapters 1 and 2 we are concerned with background material. We provide results on the classical theory of Volterra equations in Chapter 1 and on numerical methods in Chapter 2. The original material is contained in Chapters 3, 4 and 5. Here, stability results which involve the construction and analysis of finite-term recurrence relations are presented. The techniques relate to the treatment of Volterra integral and integro-differential equations. They permit the analysis of classical and 7-modified numerical methods. The results presented should be viewed as a contribution towards an understanding of numerical stability for the methods considered. The area is one in which further work (subsequent to the present investigation and involving advanced techniques) has been performed and where open questions still remain. The techniques which are employed in this thesis are applicable in other areas of numerical analysis and therefore have intrinsic interest.

Boundary value methods for Volterra integral and integro-differential equations

We consider a fully practical finite element approximation of the nonlinear parabolic Cahn–Hilliard system urn:x-wiley:0749159X:media:num21972:num21972-math-0001 urn:x-wiley:0749159X:media:num21972:num21972-math-0002 subject to an initial condition on the conserved order parameter , and mixed boundary conditions. Here, is the interfacial parameter, is the field strength parameter, is the obstacle potential, is the diffusion coefficient, and denotes differentiation with respect to the second argument. Furthermore, w is the chemical potential and is the electrostatic potential. The system, in the context of nanostructure patterning, has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field. In the limit , it reduces to a sharp interface problem that models the evolution of an unstable interface between two dielectric media in the presence of a quasistatic electric field. On introducing a finite element approximation for the above Cahn–Hilliard system, we prove existence and stability of a discrete solution. Moreover, in the case of two space dimensions, we are able to prove convergence and hence existence of a solution to the considered system of partial differential equations. We demonstrate the practicality of our finite element approximation with several numerical simulations in two and three space dimensions. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1890–1924, 2015

Resolvent conditions for discretizations of diffusion-convection-reaction equations in several space dimensions

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problem between its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, and tends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction with one's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension, oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.

We prove the convergence of certain second-order numerical methods to weak solutions of the Navier–Stokes equations satisfying, in addition, the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli–Kohn–Nirenberg. More precisely, we treat the space-periodic case in three space dimensions and consider a full discretization in which the classical Crank–Nicolson method (θ-method with theta =1/2) is used to discretize the time variable. In contrast, in the space variables, we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions toward weak solutions (satisfying a precise local energy balance) without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes, providing alternate proofs of the existence of “physically relevant” solutions in three space dimensions.

Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. In this paper, we propose an efficient and accurate numerical method for the one and two dimensional nonlinear viscous Burgers’ equations and coupled viscous Burgers’ equations with various values of viscosity subject to suitable initial and boundary conditions. The method is based on the Chebyshev collocation technique in space and the fourth-order Runge–Kutta method in time. This proposed scheme is robust, fast, flexible, and easy to implement using modern mathematical software such as Matlab. Furthermore, it can be easily modified to handle other problems. Extensive numerical simulations are presented to demonstrate the accuracy of the proposed scheme. The numerical solutions for different values of the Reynolds number are compared with analytical solutions as well as other numerical methods available in the literature. Compared to other numerical methods, the proposed method is shown to have higher accuracy with fewer nodes. Our numerical experiments show that the Chebyshev collocation method is an efficient and reliable scheme for solving Burgers’ equations with reasonably high Reynolds number.

Difference scheme for two-phase flow

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADER-DG methods on collocated grids for compressible fluids (2) spectral semi-implicit and spectral space–time DG methods on edge-based staggered grids for the incompressible Navier–Stokes equations. In this work, the high-resolution properties of the aforementioned numerical methods are significantly enhanced within a 'cell-by-cell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADER-DG is a novel, communication-avoiding family of algorithms, which achieves high order of accuracy in time not via the standard multi-stage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an element-local predictor stage. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. In the ADER-DG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magneto-hydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand. In this context, we derived two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor $$\theta \in [0.5,1]$$ for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to $$N = 12$$, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.

Gegenstand dieser Arbeit ist die numerische Analysis von Optimierungsproblemen mit partiellen Differentialgleichungen (PDEs), deren Zustand oder Steuerung punktweisen Ungleichungsbeschränkungen unterliegt. Wir interessieren uns insbesondere für nichtkonvexe Probleme mit semilinearer Zustandsgleichung. Es liegt in der Natur derartiger Probleme, dass eine Lösung oftmals nur numerisch gefunden werden kann. Man interessiert sich deshalb für den Fehler zwischen einer (lokalen) Lösung des kontinuierlichen Problems und einer zugehörigen (lokalen) diskreten Lösung. Eine umfassende Diskussion des kontinuierliche Problems ist dabei eine Grundvoraussetzung. Insbesondere bei vorhanden punktweisen Zustandsbeschränkungen treten spezifische Schwierigkeiten sowohl analytischer als auch numerischer Art auf, denen man entweder direkt oder mit Hilfe von Regularisierungsansätzen begegnen kann. Mit dieser Dissertation leisten wir auf verschiedene Weise neue Beiträge zur Diskussion von Optimalsteuerungsproblemen mit punktweisen Zustandsschranken aber auch reinen Kontrollschranken. Nach kurzer Einführung in die Thematik und Bereitstellung gewisser Grundlagen beschäftigen wir uns in Kapitel 3 mit einem elliptischen semiinfiniten Optimalsteuerungsproblem. Bekannte Resultate zu notwendigen und hinreichenden Optimalitätsbedingungen, die vergleichsweise hohe Regularität von Lösungen elliptischer PDEs und die Endlichdimensionalität des Steuerungsraumes lassen eine direkte Diskussion von a priori Diskretisierungsfehlerabschätzungen für dieses Problem zu. Unser Hauptergebnis in diesem Kapitel ist eine a priori Fehlerschranke der Ordnung O(h^2|ln h|) für lokale Lösungen einer Finite-Elemente-Diskretisierung des Optimalsteuerungsproblems mit Gitterweite h in einem zweidimensionalen Ortsgebiet. Dazu stellen wir gewisse Annahmen an die Struktur der aktiven Menge. In Kapitel 4 betrachten wir ein parabolisches Optimalsteuerungsproblem mit punktweise beschräkten Steuerungsfunktionen, semilinearer Zustandsgleichung und punktweisen Zustandsbeschränkungen im gesamten Orts-Zeit-Gebiet. Im Gegensatz zu dem in Kapitel 3 diskutierten elliptischen Problem sind hier u.a. hinreichende Bedingungen zweiter Ordnung nur für eindimensionale Ortsgebiete verfügbar. Auch läßt die Verwendung von Steuerungsfunktionen an Stelle endlich vieler Parameter keine sinnvollen a priori Annahmen an die Struktur der aktiven Menge zu. Wir regularisieren daher das Problem mit der auf Meyer, Rösch und Tröltzsch zurückgehenden Lavrentievregularisierung, und können so unter anderem auf bekannte Resultate zurückgreifen, die eine höhere Regularität der Lagrangeschen Multiplikatoren sichern und eine tiefergehende Analysis ermöglichen. Wir beweisen ein Konvergenzresultat für lokale Lösungen des regularisierten Problems und weisen die lokale Eindeutigkeit regularisierter Lösungen nach. In Kapitel 5 untersuchen wir die Finite-Element-Diskretisierung eines kontrollbeschränkten parabolischen Optimalsteuerungsproblems mit semilinearer Zustandsgleichung. Wir beweisen Fehlerordnungen für diskrete lokale Lösungen in der L^2-Norm. Dabei erweitern wir Resultate, die für linear-quadratische Probleme bekannt sind, auf den nichtkonvexen Fall. Es müssen insbesondere Beschränktheitsresultate in der L^infty-Norm der semidiskreten und diskreten Zustände gezeigt werden, die unabhängig von den Diskretisierungsparametern gelten. Außerdem erfordert die Diskussion lokal optimaler Lösungen, dass Konvergenzresultate und quadratische Wachstumsbedingungen in denselben Normen betrachtet werden.

A set of two numerical methods is presented for determining optimum configuration of the poloidal field coils of a large tokamak. The first method is an application of the virtual-casing principle developed by Zakharov and determines both the currents and the positions of a system of the coils. The second one used a simplex method of non-linear programming for optimization calculations of the coil positions in a given area in the two dimensional space. This is needed to meet various practical requirements in the design of tokamaks. These methods have been applied in the design of JT-60 (JAERI large tokamak) to determine the positions of the ohmic heating coils, and shown to be effective when they are used in combination.