Showing papers on "Partial differential equation published in 1999"

Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction

Abstract: The Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014 linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014 State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013 Order and TVD Methods for Scalar Equations.- High#x2013 Order and TVD Schemes for Non#x2013 Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013 Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.

A penalization method to take into account obstacles in incompressible viscous flows

Abstract: From the Navier-Stokes/Brinkman model, a penalization method has been derived by several authors to compute incompressible Navier-Stokes equations around obstacles. In this paper, convergence theorems and error estimates are derived for two kinds of penalization. The first one corresponds to $L^2$ penalization inducing a Darcy equation in the solid body, the second one corresponds to a $H^1$ penalization and induces a Brinkman equation in the body. Numerical tests are performed to confirm the efficiency and accuracy of the method.

Mixed quantum-classical dynamics

Abstract: Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.

The nature of mathematical modeling

Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.

Symmetries and Conservation Laws for Differential Equations of Mathematical Physics

Abstract: Ordinary differential equations First-order equations The theory of classical symmetries Higher symmetries Conservation laws Nonlocal symmetries From symmetries of partial differential equations towards secondary (""quantized"") calculus Bibliography Index.

Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations

Abstract: Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter, from the rarefied kinetic regimes to the hydrodynamic diffusive regimes. An earlier approach in [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] reformulates such systems into the common hyperbolic relaxation system by Jin and Xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time-splitting method to solve the relaxation system. Here we observe that the combination of the two time-split steps may yield hyperbolic-parabolic systems that are more advantageous, in both stability and efficiency, for numerical computations. We show that such an approach yields a class of asymptotic-preserving (AP) schemes which are suitable for the computation of multiscale kinetic problems. We use the Goldstein--Taylor and Carleman models to illustrate this approach.

Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s

Abstract: We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.

The nature of mathematical modeling

Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.

Multicomponent Flow Modeling

Abstract: We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations. We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves, also termed anchored waves. We further discuss related models which have a similar hyperbolic-parabolic structure, notably the Saint-Venant system with a temperature equation as well as the equations governing chemical equilibrium flows. We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure. We finally discuss numerical algorithms specifically devoted to complex chemistry flows, in particular the evaluation of multicomponent transport properties, as well as the impact of multicomponent transport.

Numerical Method of Lines

Abstract: The method of lines for partial differential equations consists in replacing spatial derivatives by difference expressions. Then the partial equation is transformed into a system of ordinary differential equations. The method is used for approximation of solutions of nonlinear differential problems of parabolic type by solutions of ordinary equations ([91, 153, 219, 220, 222, 225, 238]). The method is also treated as a tool for proving of existence theorems for differential problems corresponding to parabolic equations [223, 224, 227] or first-order hyperbolic systems [101, 157]. Simple examples of the method of lines for nonlinear functional differential equations were considered in [29, 108, 128]. The method for equations of higher orders is studied in [91]. The book [189] demonstrates lots of examples of the use of the numerical method of lines. Convergence analysis of one step difference methods generated by the numerical method of lines was investigated in [186].

Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method

Abstract: An efficient semi-implicit Fourier spectral method is implemented to solve the Cahn-Hilliard equation with a variable mobility. The method is orders of magnitude more efficient than the conventional forward Euler finite-difference method, thus allowing us to simulate large systems for longer times. We studied the coarsening kinetics of interconnected two-phase mixtures using a Cahn-Hilliard equation with its mobility depending on local compositions. In particular, we compared the kinetics of bulk-diffusion-dominated and interface-diffusion-dominated coarsening in two-phase systems. Results are compared with existing theories and previous computer simulations.

On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity

Abstract: The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach.

A Simple Method for Compressible Multifluid Flows

Abstract: A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunov-type scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 10 5). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer--Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes.

Derivation, properties, and simulation of a gas-kinetic-based, nonlocal traffic model

Abstract: We derive macroscopic traffic equations from specific gas-kinetic equations, dropping some of the assumptions and approximations made in previous papers. The resulting partial differential equations for the vehicle density and average velocity contain a non-local interaction term which is very favorable for a fast and robust numerical integration, so that several thousand freeway kilometers can be simulated in real-time. The model parameters can be easily calibrated by means of empirical data. They are directly related to the quantities characterizing individual driver-vehicle behavior, and their optimal values have the expected order of magnitude. Therefore, they allow to investigate the influences of varying street and weather conditions or freeway control measures. Simulation results for realistic model parameters are in good agreement with the diverse non-linear dynamical phenomena observed in freeway traffic.

Fluid Flow Phenomena: A Numerical Toolkit

Abstract: This book deals with the simulation of the incompressible Navier-Stokes equations for laminar and turbulent flows. The book is limited to explaining and employing the finite difference method. It furnishes a large number of source codes which permit to play with the Navier-Stokes equations and to understand the complex physics related to fluid mechanics. Numerical simulations are useful tools to understand the complexity of the flows, which often is difficult to derive from laboratory experiments. This book, then, can be very useful to scholars doing laboratory experiments, since they often do not have extra time to study the large variety of numerical methods; furthermore they cannot spend more time in transferring one of the methods into a computer language. By means of numerical simulations, for example, insights into the vorticity field can be obtained which are difficult to obtain by measurements. This book can be used by graduate as well as undergraduate students while reading books on theoretical fluid mechanics; it teaches how to simulate the dynamics of flow fields on personal computers. This will provide a better way of understanding the theory. Two chapters on Large Eddy Simulations have been included, since this is a methodology that in the near future will allow more universal turbulence models for practical applications. The direct simulation of the Navier-Stokes equations (DNS) is simple by finite-differences, that are satisfactory to reproduce the dynamics of turbulent flows. A large part of the book is devoted to the study of homogeneous and wall turbulent flows. In the second chapter the elementary concept of finite difference is given to solve parabolic and elliptical partial differential equations. In successive chapters the 1D, 2D, and 3D Navier-Stokes equations are solved in Cartesian and cylindrical coordinates. Finally, Large Eddy Simulations are performed to check the importance of the subgrid scale models. Results for turbulent and laminar flows are discussed, with particular emphasis on vortex dynamics. This volume will be of interest to graduate students and researchers wanting to compare experiments and numerical simulations, and to workers in the mechanical and aeronautic industries.

Advanced topics in computational partial differential equations : numerical methods and Diffpack programming

Abstract: X. Cai, E. Acklam, H. P. Langtangen, A. Tveito: Parallel Computing.- X. Cai: Overlapping Domain Decomposition Methods.- K.-A. Mardal, G. W. Zumbusch, H. P. Langtangen: Software Tools for Multigrid Methods.- K.-A. Mardal, H. P. Langtangen: Mixed Finite Elements.- K.-A. Mardal, J. Sundnes, H. P. Langtangen, A. Tveito: Systems of PDEs and Block Preconditioning.- A. Odegard, H. P. Langtangen, A. Tveito: Object-Oriented Implementation of Fully Implicit Methods for Systems of PDEs.- H. P. Langtangen, H. Osnes: Stochastic Partial Differential Equations.- H. P. Langtangen and K.-A. Mardal: Using Diffpack from Python Scripts.- X. Cai, A. M. Bruaset, H. P. Langtangen, G. T. Lines, K. Samuelsson, W. Shen, A. Tveito, G. Zumbusch: Performance Modeling of PDE Solvers.- J. Sundnes, G.T. Lines, P. Grottum, A. Tveito: Numerical Methods and Software for Modeling the Electrical Activity in the Human Heart.- O. Skavhaug, B. F. Nielsen, A. Tveito: Mathematical Models of Financial Derivatives.- O. Skavhaug, B. F. Nielsen, A. Tveito: Numerical Methods for Financial Derivatives.- T. Thorvaldsen, H. P. Langtangen, H. Osnes: Finite Element Modeling of Elastic Structures.- K. M. Okstad, T. Kvamsdal: Simulation of Aluminum Extrusion.- A. Kjeldstad, H. P. Langtangen, J. Skogseid, K. Bjorlykke: Simulation of Deformations, Fluid Flow and HeatTransfer in Sedimentary Basins

BSDEs, weak convergence and homogenization of semilinear PDEs

Abstract: In these lectures, we present the theory of backward stochastic differential equations, and its connection with solutions of semilinear second order partial differential equations of parabolic and elliptic type. This connection provides a probabilistic tool for studying solutions of semilinear PDEs. We apply our results to the proof of the homogenization result for such PDEs, both with periodic and random coefficients. For that purpose, we need to present the theory of weak limits of solutions of backward stochastic differential equations. We also present a complete probabilistic proof, under apparently minimal assumptions, of the homogenization result of linear second order PDEs.

Asymptotic model of electroporation

Abstract: Electroporation is described mathematically by a partial differential equation (PDE) that governs the distribution of pores as a function of their radius and time. This PDE does not have an analytical solution and, because of the presence of disparate spatial and temporal scales, numerical solutions are hard to obtain. These difficulties limit the application of the PDE only to experimental setups with a uniformly polarized membrane. This study performs a rigorous, asymptotic reduction of the PDE to an ordinary differential equation (ODE) that describes the dynamics of the pore density $N(t).$ Given $N(t),$ the precise distribution of the pores in the space of their radii can be determined by an asymptotic approximation. Thus, the asymptotic ODE represents most of the phenomenology contained in the PDE. It is easy to solve numerically, which makes it a powerful tool to study electroporation in experimental setups with significant spatial dependence, such vesicles or cells in an external field.

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

Abstract: We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c≠0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c≠0. Convergence results for solutions are obtained at the singular perturbation limit c → 0.

Meshless Galerkin methods using radial basis functions

Abstract: We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

Abstract: We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of Lp convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.

Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

Abstract: From the Publisher: The target audience of this book is students and researchers in computational sciences who need to develop computer codes for solving partial differential equations. The exposition is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods and basic finite difference methods from a computational point of view. The main emphasis regards development of flexible computer programs, using the numerical library Diffpack. The application of Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. Diffpack is a modern software development environment based on C++ and object-oriented programming.

Wavelet and Fourier Methods for Solving the Sideways Heat Equation

Abstract: We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x=1, where the solution is wanted for $0 \leq x < 1$. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge--Kutta method. We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

An advanced algorithm for solving partial differential equation in cardiac conduction

Abstract: An advanced integration method for solving reaction-diffusion-type equations for cardiac conduction is suggested. Operator splitting and adaptive time step methods were used in this method, which can significantly speed up integration while preserving accuracy.

A Second-Order Rosenbrock Method Applied to Photochemical Dispersion Problems

Abstract: A second-order, L-stable Rosenbrock method from the field of stiff ordinary differential equations is studied for application to atmospheric dispersion problems describing photochemistry, advective, and turbulent diffusive transport. Partial differential equation problems of this type occur in the field of air pollution modeling. The focal point of the paper is to examine the Rosenbrock method for reliable and efficient use as an atmospheric chemical kinetics box-model solver within Strang-type operator splitting. In addition, two W-method versions of the Rosenbrock method are discussed. These versions use an inexact Jacobian matrix and are meant to provide alternatives for Strang-splitting. Another alternative for Strang-splitting is a technique based on so-called source-splitting. This technique is briefly discussed.