Operator-splitting methods respecting Eigenvalue problems for convection-diffusion and wave equations

We discuss iterative operator splitting method for wave-equations motivated from the eigenvalue-problem to decide the splitting process. The operator-splitting methods are wel-know to solve such complicated multidimensional and multi physics problems. Often it is not good understood how to decouple the underlying operators. We propose a method based on computing the eigenvalues for the simpler problem to decide the splitting operators and the time-steps. We present the analysis and the numerical results

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

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

Iterative operator-splitting methods for linear problems

The operator-splitting methods base on splitting of the complex problem into the sequence of the simpler tasks. A useful method is the iterative splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of splitting the time interval and the traditional iterative operator splitting. We analyze the local splitting error of the method. Numerical examples are given in order to demonstrate the method

Operator-Splitting Methods Respecting Eigenvalue Problems for Nonlinear Equations and Applications for Burgers Equations

In this article we consider iterative operator-splitting methods for nonlinear differential equations with respect to their eigenvalues. The main feature of the proposed idea is the fixed-point iterative scheme that linearizes our underlying equations. Based on the approximated eigenvalues of such linearized systems we choose the order of the the operators for our iterative splitting scheme. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes

Iterative operator-splitting methods for nonlinear differential equations and applications of deposition processes

In this article we consider iterative operator-splitting methods for nonlinear differential equations. The main feature of the proposed idea is the embedding of Newton's method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes

Iterative Operator-Splitting Methods with higher order Time-Integration Methods and Applications for Parabolic Partial Differential Equations

In this paper we design higher order time integrators for systems of stiff ordinary differential equations. We could combine implicit Runge-Kutta- and BDF-methods with iterative operator-splitting methods to obtain higher order methods. The motivation of decoupling each complicate operator in simpler operators with an adapted time-scale allow us to solve more efficiently our problems. We compare our new methods with the higher order Fractional-Stepping Runge-Kutta methods, developed for stiff ordinary differential equations. The benefit will be the individual handling of each operators with adapted standard higher order time-integrators. The methods are applied to convection-diffusion-reaction equations and we could obtain higher order results. Finally we discuss the iterative operator-splitting methods for the applications to multi-physical problems

Fractional-splitting and domain-decomposition methods for parabolic problems and applications

In this paper we consider the first order fractional splitting method to solve decomposed complex equations with multi-physical processes for applications in porous media and phase-transitions. The first order fractional splitting method is also considered as basic solution for the overlapping Schwarz-Waveform-Relaxation method for an overlapped subdomains. The accuracy and the efficiency of the methods are investigated through the solution of different model problems of scalar, coupling and decoupling systems of convection reaction diffusion equation