TY - GEN A1 - Nägel, Arne A1 - Deuflhard, Peter A1 - Wittum, Gabriel T1 - Efficient Stiff Integration of Density Driven Flow Problems N2 - The paper investigates the efficient use of a linearly implicit stiff integrator for the numerical solution of density driven flow problems. Upon choosing a one-step method of extrapolation type (code LIMEX), the use of full Jacobians and reduced approximations are discussed. Numerical experiments include nonlinear density flow problems such as diffusion from a salt dome (2D), a (modified) Elder problem (3D), the saltpool benchmark (3D) and a real life salt dome problem (2D). The arising linear equations are solved using either a multigrid preconditioner from the software package UG4 or the sparse matrix solver SuperLU. T3 - ZIB-Report - 18-54 KW - stiff integration for PDEs KW - linearly implicit Euler discretization Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-70891 SN - 1438-0064 ER - TY - GEN A1 - Fischer, Lisa A1 - Götschel, Sebastian A1 - Weiser, Martin T1 - Lossy data compression reduces communication time in hybrid time-parallel integrators N2 - Parallel in time methods for solving initial value problems are a means to increase the parallelism of numerical simulations. Hybrid parareal schemes interleaving the parallel in time iteration with an iterative solution of the individual time steps are among the most efficient methods for general nonlinear problems. Despite the hiding of communication time behind computation, communication has in certain situations a significant impact on the total runtime. Here we present strict, yet no sharp, error bounds for hybrid parareal methods with inexact communication due to lossy data compression, and derive theoretical estimates of the impact of compression on parallel efficiency of the algorithms. These and some computational experiments suggest that compression is a viable method to make hybrid parareal schemes robust with respect to low bandwidth setups. T3 - ZIB-Report - 17-25 KW - parallel-in-time integration KW - hybrid parareal KW - convergence KW - lossy compression KW - MPI Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-63961 SN - 1438-0064 ER - TY - GEN A1 - Ruprecht, Daniel A1 - Schädle, Achim A1 - Schmidt, Frank A1 - Zschiedrich, Lin T1 - Transparent boundary conditons for time-dependent problems N2 - A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients. T3 - ZIB-Report - 07-12 KW - transparent boundary condition KW - non-reflecting boundary condition KW - pole condition KW - wave equation KW - Schrödinger equation KW - drift diffusion equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9558 ER - TY - GEN A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Palencia, Cesar A1 - Schädle, Achim T1 - Fast Runge-Kutta approximation of inhomogeneous parabolic equations N2 - The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\varepsilon$, by solving only $$O\Big(\log N\, \log \frac1\varepsilon \Big) $$ linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm. T3 - ZIB-Report - 05-10 KW - parabolic equation KW - Runge-Kutta methods Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8443 ER - TY - GEN A1 - Nowak, Ulrich A1 - Grah, Aleksander A1 - Schreier, Manfred T1 - Parameter Estimation and Accuracy Matching Strategies for 2-D Reactor Models N2 - The mathematical modeling of a special modular catalytic reactor kit leads to a system of partial differential equation in two space dimensions. As customary, this model contains unconfident physical parameters, which may be adapted to fit experimental data. To solve this nonlinear least squares problem we apply a damped Gauss-Newton method. A method of lines approach is used to evaluate the associated model equations. By an a priori spatial discretization a large DAE system is derived and integrated with an adaptive, linearly-implicit extrapolation method. For sensitivity evaluation we apply an internal numerical differentiation technique, which reuses linear algebra information from the model integration. In order not to interfere the control of the Gauss-Newton iteration these computations are done usually very accurately and, therefore, very costly. To overcome this difficulty, we discuss several accuracy adaptation strategies, e.g., a master-slave mode. Finally, we present some numerical experiments. T3 - ZIB-Report - 03-52 KW - parameter estimation KW - accuracy matching KW - method of lines KW - differential-algebraic systems KW - extrapolation methods KW - catalytic reactor models Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7749 ER - TY - GEN A1 - Hohage, Thorsten A1 - Schmidt, Frank T1 - On the Numerical Solution of Nonlinear Schrödinger type equations in fiber optics N2 - The aim of this paper is to develop fast methods for the solution of nonlinear Schrödinger type equations in fiber optics. Using the method of lines we have to solve a stiff system of ordinary differential equations where the eigenvalues of the Jacobian are close to the imaginary axis. This is usually done by a Split Step method. Here we consider the extrapolation of Split Step methods with adaptive order and step size control. For more complicated nonlinearities, in particular stimulated Raman scattering, Split Step methods are less efficient since symmetry is either destroyed or requires much additional effort. In this case we use implicit Runge Kutta formulas of Gauß type. The key point for the efficient implementation of these methods is that the system of nonlinear algebraic equations can be solved without setting up the Jacobian. The proposed methods are compared to other methods, in particular exponential integrators, the method of Marcuse, and the method of Blow and Wood. T3 - ZIB-Report - 02-04 KW - nonlinear Schroedinger equation KW - split step methods KW - extrapolation KW - collocation methods Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6715 ER -