Discretization methods with analytical solutions for a convection-reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection-reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [Godunov 1959] and [Leveque 2002], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method. The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order TVD-methods, see [Harten 1983]. In this article we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods. For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results

Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Analytical Solutions for Convection-Diffusion-Dispersion-Reaction-Equations with Different Retardation-Factors and Applications in 2d and 3d

Our motivation to this paper came from a model simulating a wastedisposal embedded in an overlying rock. The main problem for our model are the large scales that occurred due the coupled reaction terms of our underlying system of convection-diffusion-dispersion-reactionequations. The developed methods allowed a computation over a large simulation period of more than 10000 years. Therefore we construct discretization methods of higher order, which allow large-time-steps without loss of accuracy. Based on operator-splitting methods we decouple the complex equations in simpler equations and use adequate methods to solve each equation separately. For the explicit parts that are the convection-reaction-equations we use finite-volume methods based on flux-methods with embedded analytical solutions. Whereas for the implicit parts that are the diffusion-dispersion-equations we use finitevolume methods with central discretizations. We analyze the splittingerror and the discretization error for our methods. The main part of the paper consists of the applications of our methods done with our underlying program-tool R3T. We introduced the main concepts of the program-tool that is based on the software-toolbox UG. The testexamples and benchmark problems for verifying our discretization- and solver-methods with respect to the physical behavior are presented. The benchmark-problems are the test for different material-parameters and confirm the valuation of the methods. Based on the verification of our test-problem we present the realistic model-problem of a waste-disposal in 2d with large decay-chains reacted and transported in a porous media with an underlying flowing groundwater. For the prediction of possible waste-disposals a computation with different located waste-locations is discussed. The parallel resources for the computations are presented in the case of the forced simulation-times.Peer Reviewe

An integral equation formulation for 2D steady-state advection-diffusion-reaction problems with variable coefficients

Este trabalho apresenta uma formulação de equação integral de contorno e domínio para problemas de advecção-difusão-reação com coeficientes variáveis e termo fonte. A formulação usa uma versão da solução fundamental que evita overflow numérico dos termos exponenciais e underflow dos termos em função de Bessel, para qualquer número de Péclet e qualquer tamanho de domínio. Os coeficientes usados na solução fundamental são os coeficientes locais da equação diferencial, afim de minimizar a contribuição do domínio no problema. A formulação é aplicada sem modificações para problemas puramente difusivos ou de difusão-reação. A equação integral é discretizada usando o método dos elementos de contorno, com elementos de contorno contínuos e células de domínio descontínuas. O método é validado com cinco problemas de benchmark que possuem soluções analíticas, apresentando um erro NRMSD abaixo de 1% para malhas com 1348 graus de liberdade, em todos os casos. A metodologia é usada para o estudo de dois problemas práticos. O primeiro é o problema de Graetz-Nusselt adimensional para Pe = f0; 1; 5; 10g. O segundo é um problema de pluma de dispersão de poluentes para uma fonte pontual em escoamentos de camada limite atmosférica neutramente estratificada.This work presents a boundary-domain integral equation formulation for advection-diffusionreaction problems with variable coefficients and source term. The formulation uses a version of the fundamental solution that avoids numerical overflow of the exponential term and underflow of the Bessel term, for any Péclet number and domain size. Furthermore, the coefficients used in the fundamental solution are the local coefficients of the differential equation, in order to minimize the domain contribution for the problem. The formulation is applied as-is for purely diffusive or diffusion-reaction problems. The integral equation is discretized using the boundary element method with continuous boundary elements and discontinuous domain cells. The scheme is validated against five benchmark problems with analytical solutions, presenting a NRMSD error under 1% for meshes with 1348 degrees of freedom, in all cases. The methodology is used to study two practical problems. The first is the dimensionless Graetz-Nusselt problem for Pe = f0; 1; 5; 10g. The second is the pollutant dispersion plume for a point source in neutrally stratified atmospheric boundary layer flows