High resolution compact implicit numerical scheme for conservation laws

We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.Comment: Significantly revised version that is accepted to AMC journa

Second order characteristic based schemes for chemotaxis system

Second order characteristic based schemes for the first equation of the cheomtaxis system are presented. The derived schemes belong to the finite volume methods in which the data points are the cell centres. The solutions are approximated by piecewise linear functions in 1D while piecewise bilinear functions are used in 2D case. The schemes are shown to be mass conservative. The consistency and stability are discussed analytically. Numerical experiments are provided to back up the analysis. Some directions for future work is posted

Linear and Non-linear Monotone Methods for Valuing Financial Options Under Two-Factor, Jump-Diffusion Models

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value

B-Spline Collocation Methods For Coupled Nonlinear Schrödinger Equation

In this study, the Coupled Nonlinear Schrödinger Equation (CNLSE) which models the propagation of light waves in optical fiber is solved using numerical methods namely Finite Difference Method (FDM) and B-Spline collocation methods. The equation was discretized in space and time. We propose the discretization of the nonlinear terms in the CNLSE following the Taylor approach and a newly developed approach called Besse. The theta-weighted method is used to generalize the scheme whereby the Crank-Nicolson scheme (i.e θ = 0.5) is chosen. The time derivatives are discretized by forward difference approximation. For each approach, the space dimension is then discretized by five different collocation methods independently. The first method for Taylor approach is based on FDM whereby the space derivatives are replaced by central difference approximation

Lattice Boltzmann simulations of soft matter systems

This article concerns numerical simulations of the dynamics of particles immersed in a continuum solvent. As prototypical systems, we consider colloidal dispersions of spherical particles and solutions of uncharged polymers. After a brief explanation of the concept of hydrodynamic interactions, we give a general overview over the various simulation methods that have been developed to cope with the resulting computational problems. We then focus on the approach we have developed, which couples a system of particles to a lattice Boltzmann model representing the solvent degrees of freedom. The standard D3Q19 lattice Boltzmann model is derived and explained in depth, followed by a detailed discussion of complementary methods for the coupling of solvent and solute. Colloidal dispersions are best described in terms of extended particles with appropriate boundary conditions at the surfaces, while particles with internal degrees of freedom are easier to simulate as an arrangement of mass points with frictional coupling to the solvent. In both cases, particular care has been taken to simulate thermal fluctuations in a consistent way. The usefulness of this methodology is illustrated by studies from our own research, where the dynamics of colloidal and polymeric systems has been investigated in both equilibrium and nonequilibrium situations.Comment: Review article, submitted to Advances in Polymer Science. 16 figures, 76 page

Robust computational methods to simulate slow-fast dynamical systems governed by predator-prey models

Philosophiae Doctor - PhDNumerical approximations of multiscale problems of important applications in ecology are investigated. One of the class of models considered in this work are singularly perturbed (slow-fast) predator-prey systems which are characterized by the presence of a very small positive parameter representing the separation of time-scales between the fast and slow dynamics. Solution of such problems involve multiple scale phenomenon characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations, which are typically challenging to approximate numerically. Granted with a priori knowledge, various time-stepping methods are developed within the framework of partitioning the full problem into fast and slow components, and then numerically treating each component differently according to their time-scales. Nonlinearities that arise as a result of the application of the implicit parts of such schemes are treated by using iterative algorithms, which are known for their superlinear convergence, such as the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed point methods

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248

Finite Difference Computing with Exponential Decay Models

Computational Science and Engineering; software Engineering; Programming Technique

Determination of the time-dependent thermal grooving coefficient

Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finitedifference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data