An iterated pseudospectral method for delay partial differential equations

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

Coal fired power plants contribute significantly to greenhouse gas emission, notably CO2. A novel method to effectively reduce the amount of CO2 emission is to cofire a high fraction of biomass and coal at oxygen enriched environments. However, this technique has not been demonstrated on a large-scale yet. The objective of this PhD research is to perform a detailed modeling study of combustion of a single biomass particle and to establish reduced models which can be used for describing the main characteristics of pyrolyzing and combusting single biomass particles to be employed in design codes of industrial furnaces. To accomplish the goals of this PhD thesis, the research has been carried out in two main stages. First, the sub-processes involved in the biomass combustion are identified and described with a one-dimensional mathematical model based on conservation of mass, energy and momentum. The model encompasses the kinetics of biomass pyrolysis, homogeneous reactions and heterogeneous char oxidation and gasification reactions, coupled with transient transport equations. Subsequently, this model is implemented in an in-house code and a comprehensive numerical study on pyrolysis and combustion of single biomass particles is conducted. The accuracy of the model is examined by comparing its predictions with several experimental data obtained from the literature on pyrolysis and combustion of various types of single biomass particles. The computer code based on the detailed model allows one to observe time and space evolution of several parameters including biomass and char densities, gaseous species mass fractions, porosity, internal pressure, mass flux of volatiles within the pores of the solid matrix, and temperature. The model is used for simulation of combustion of particles of three common shapes; i.e. slab, cylinder and sphere. The results of the detailed modeling study reveal that the combustion of a single biomass particle at the conditions of industrial furnaces (small particles and high heating conditions) consists of three main sub-processes: preheating, pyrolysis, and char oxidation. Therefore, any simplified/reduced particle model should account for these three processes. In the next stage of the project, simplified models are developed to predict the main characteristics of pyrolyzing and combusting single biomass particles. Initially, the preheating stage is modeled using a time and space integral method, which allows one to convert the partial differential form of the heat transfer equation into an algebraic equation. This treatment is then applied to model the pyrolysis process. Two possible regimes are identified: thermally thin and thermally thick particles. A model is established for both regimes, which consists of a set of algebraic equations. This treatment highly simplifies the pyrolysis model so that it can be used in practical applications, which may involve thousands or even millions of particles. The validation of the simplified preheating and pyrolysis models is carried out using various experiments and the results of the detailed model based on partial differential equations (PDEs). The char particle oxidation and gasification processes as the last stages of the particle combustion process, are modeled using the shrinking core approximation. The accuracy of this model is assessed using the experiments reported in past studies. The model is used to study the dynamics of biomass char combustion at oxy-fuel conditions. The effects of the main process parameters on the maximum particle temperature and burnout time are examined. It is found that oxy-fuel combustion with an oxygen mass fraction of 0.3 and higher may lead to a considerable reduction in particle temperature and burnout time compared to the conventional operating case with air as the gasifying agent. In the last stage of this PhD research, the simplified models of biomass particle pyrolysis and char combustion are combined to establish a reduced model for combustion of a single biomass particle. The accuracy of this simplified combustion model is examined using the measured data available in the literature as well as the results of the detailed model. As a conclusion, the simplified models developed in this thesis for pyrolysis and combustion of single biomass particles are efficient enough to capture the main process parameters, and computationally cheaper than the PDE -based models so that they can be used in the design codes of biomass furnaces.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

The presence of noise in images is unavoidable.It may be formed by image formation process,image recording,image transmission etc.These random distortions make it difficult to perform any required image processing task.High quality noise free remote sensing images are necessary for various applications.Therefore, removal of noise is necessary. The purpose of image denoising is to preserve edges as far as possible while removing noise, making the resulting images approximate the ideal image.This paper proposes a method for denoising remote sensing images using a combination of second order and fourth order partial differential equations.The advantages of both second order and fourth order partial differential equations are utilized here.The image is denoised using second order partial differential equations , fourth order partial differential equations and the combination of both.A comparison of the three methods is also presented.The proposed algorithm smooth out more noise and conserve more detail in the denoising process.

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

In this paper the effect of axial load on dynamic behavior of a simple frame, subjected to harmonic, seismic and earthquake excitation is investigated. The equations of motion are considered for two types of small and large deformations. The method of multiple scales is applied to solve the differential equations of motion with harmonic loading and for small and large deformations. Then, the steady state response near one-to-one resonance condition is studied. The results show that the dynamic behavior of the frame under axial load is completely different in resonance and non-resonance cases. The equations of motion with earthquake loading is also considered and the effect of axial load in the frame behavior under the time history and the response spectrum of the model is studied. Although white noise as a stochastic loading is applied to the model and, the results are approximated using the method of stochastic differential equations so, the mean value and covariance are calculated and the effect of axial force on them is investigated.

The reversible Hopf bifurcation with 1:1 resonance holds the key to the presence of spatially localized steady states in many partial differential equations on the real line. Two different techniques for computing the normal form for this bifurcation are described and applied to the Swift-Hohenberg equation with cubic/quintic and quadratic/cubic nonlinearities.

Computational Fluid Dynamics (CFD) deals with computing the equations of fluid flows using numerical methods instead of partial differential equations. The Discrete-Event System specification (DEVS) theory has already been used to approximate various continuous systems by applying a quantized state system approach. In this research, we experiment with a new method: using Cell-DEVS theory to create a uniform set of rules for CFD to apply to each cell and execute the state changes of the cells asynchronously. This harmonized set of state changes can effectively render the fluid dynamics, by applying the accurate rules that represents the behavior of the fluid.

MHD boundary layer flow in permeable medium along vertical plate with power-law deviation wall temperature in a gyratory system for the effects of Hall and ion-slip currents have investigated numerically. The ordinary differential equations are obtained from considered system of partial differential equations using usual transform. The Nachtsheim- Swigert shooting iteration method along with sixth order Runge-Kutta integration technique has employed to solve the non-dimensional system of ordinary differential equations. The velocities and temperature profiles are displayed graphically for the influence of physical parameters. The numerical values of the rate of change of velocities and temperature have been estimated for concerned physical parameters and represented in table.

We describe an extension of previous results on degenerate abstract equations described by Favini and Yagi in their monograph cited in the references.This allows us to study degenerate differential equations with fractional derivative in time. Both direct and relative inverse problems are studied. Finally, various applications to degenerate partial differential equations are described.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

The time required to execute real-world scientific computations is a major issue. A single simulation may last hours, days, or even weeks to reach a certain level of accuracy, despite running on large-scale parallel architectures. Strict time limits may often be imposed too -- 60 minutes in the case of the UK Met Office to produce a forecast. In this thesis, it is demonstrated that by raising the level of abstraction, the performance of a class of numerical methods for solving partial differential equations is improvable with minimal user intervention or, in many circumstances, with no user intervention at all. The use of high level languages to express mathematical problems enables domain-specific optimization via compilers. These automated optimizations are proven to be effective in a variety of real-world applications and computational kernels. The focus is on numerical methods based on unstructured meshes, such as the finite element method. The loop nests for unstructured mesh traversal are often irregular (i.e., they perform non-affine memory accesses, such as A[B[i]]), which makes reordering transformations for data locality essentially impossible for low level compilers. Further, the computational kernels are often characterized by complex mathematical expressions, and manual optimization is simply not conceivable. We discuss algorithmic solutions to these problems and present tools for their automation. These tools -- the COFFEE compiler and the SLOPE library -- are currently in use in frameworks for solving partial differential equations.

Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

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