TY - GEN A1 - Fröhlich, Jochen A1 - Lang, Jens T1 - Twodimensional Cascadic Finite Element Computations of Combustion Problems N2 - We present an integrated time--space adaptive finite element method for solving systems of twodimensional nonlinear parabolic systems in complex geometry. The partial differential system is first discretized in time using a singly linearly implicit Runge--Kutta method of order three. Local time errors for the step size control are defined by an embedding strategy. These errors are used to propose a new time step by a PI controller algorithm. A multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space. The local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation. This two--fold adaptivity successfully ensures an a priori prescribed tolerance of the solution. The devised method is applied to laminar gaseous combustion and to solid--solid alloying reactions. We demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm. T3 - ZIB-Report - SC-96-05 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2167 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Schneider, Kai T1 - An adaptive Wavelet-Vaguelette Algorithm for the Solution of Nonlinear PDEs N2 - The paper describes a fast algorithm for the discrete periodic wavelet transform and its inverse without using the scaling function. The approach permits to compute the decomposition of a function into a lacunary wavelet basis, i.e. a basis constituted of a subset of all basis functions up to a certain scale, without modification. The construction is then extended to operator--adapted biorthogonal wavelets. This is relevant for the solution of non--linear evolutionary PDEs where a priori information about the significant coefficients is available. We pursue the approach described in FrSc94 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids. Here, we present an improved construction employing the cardinal function of the multiresolution. The new method is applied to the Helmholtz equation and illustrated by comparative numerical results. It is then extended for the solution of a nonlinear parabolic PDE with semi--implicit discretization in time and self--adaptive wavelet discretization in space. Results with full adaptivity of the spatial wavelet discretization are presented for a one--dimensional flame front as well as for a two--dimensional problem. T3 - ZIB-Report - SC-95-28 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1941 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Deuflhard, Peter T1 - Moving Weight Galerkin Methods for Turbulent Reactive Flows N2 - Adopting a statistical approach for the computation of turbulent combustion flows an approximation for the probability density function (PDF) of the composition variables is often required to treat the highly non-linear reaction term in a satisfactory way. One class of methods currently being used are the moment methods which employ transport equations for low order statistical moments and use a parametrized shape of the PDF. A second class solves a transport equation for the joint PDF by a Monte Carlo method. In the present paper we develop an intermediate algorithm based on a Galerkin method for the PDF transport equation. The solution is developed in terms of an orthogonal or bi-orthogonal basis of a suitable Hilbert space. The unconventional use of the related weight function as a prefactor (moving weight approach) permits adaptivity and results in a generalization of the $\beta-$closure for bounded scalar quantities. We present the approximation procedure in detail and apply it to the evolution of the composition in a homogeneous well-stirred reactor. The extension to non-homogeneous flow simulations is straightforward. T3 - ZIB-Report - SC-94-36 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1646 ER - TY - GEN A1 - Brand, Rüdiger A1 - Freeden, Willi A1 - Fröhlich, Jochen T1 - An Adaptive Hierarchical Approximation Method on the Sphere Using Axisymmetric Locally Supported Basis Functions N2 - The paper discusses the approximation of scattered data on the sphere which is one of the major tasks in geomathematics. Starting from the discretization of singular integrals on the sphere the authors devise a simple approximation method that employs locally supported spherical polynomials and does not require equidistributed grids. It is the basis for a hierarchical approximation algorithm using differently scaled basis functions, adaptivity and error control. The method is applied to two examples one of which is a digital terrain model of Australia. T3 - ZIB-Report - SC-95-02 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1682 ER - TY - GEN A1 - Fröhlich, Jochen T1 - Parameter Derivatives of the Jacobi Polynomials and the Gaussian Hypergeometric Function. N2 - The paper's main result is a simple derivation rule for the Jacobi polynomials with respect to their parameters, i.e. for $\partial_{\alpha} P_n^{\alpha,\beta}$, and $\partial_{\beta} P_n^{\alpha,\beta}$. It is obtained via relations for the Gaussian hypergeometric function concerning parameter derivatives and integer shifts in the first two arguments. These have an interest on their own for further applications to continuous and discrete orthogonal polynomials. The study is motivated by a Galerkin method with moving weight, presents all proofs in detail, and terminates in a brief discussion of the generated polynomials. T3 - ZIB-Report - SC-94-15 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1440 ER - TY - GEN A1 - Renaut, Rosemary A1 - Fröhlich, Jochen T1 - A Pseudospectral Chebychev method for the 2D Wave Equation with Domain Stretching and Absorbing Boundary Conditions N2 - In this paper we develop a method for the simulation of wave propagation on artificially bounded domains. The acoustic wave equation is solved at all points away from the boundaries by a pseudospectral Chebychev method. Absorption at the boundaries is obtained by applying one-way wave equations at the boundaries, without the use of damping layers. The theoretical reflection coefficient for the method is compared to theoretical estimates of reflection coefficients for a Fourier model of the problem. These estimates are confirmed by numerical results. Modification of the method by a transformation of the grid to allow for better resolution at the center of the grid reduces the maximum eigenvalues of the differential operator. Consequently, for stability the maximum timestep is $O(1/N)$ as compared to $O(1/N^2)$ for the standard Chebychev method. Therefore, the Chebychev method can be implemented with efficiency comparable to that of the Fourier method. Moreover, numerical results presented demonstrate the superior performance of the new method. T3 - ZIB-Report - SC-95-07 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1730 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Schneider, Kai T1 - Computation of Decaying Turbulence in an Adaptive Wavelet Basis N2 - The paper presents computations of decaying two--dimensional turbulence in an adaptive wavelet basis. At each time step the vorticity is represented by an adaptively selected set of wavelet functions which adjusts to the instantaneous distribution of vorticity. The results of this new algorithm are compared to a classical Fourier method and a Fourier method supplemented with wavelet compression in each time step. T3 - ZIB-Report - SC-96-11 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2227 ER - TY - GEN A1 - Fröhlich, Jochen A1 - Lang, Jens A1 - Roitzsch, Rainer T1 - Selfadaptive Finite Element Computations with Smooth Time Controller and Anisotropic Refinement N2 - We present Multilevel Finite Element computations for twodimensional reaction-diffusion systems modelling laminar flames. These systems are prototypes for extreme stiffness in time and space. The first of these two rather general features is accounted for by an improved control mechanism for the time step. The second one is reflected through very thin travelling reaction fronts for which we propose an anisotropic discretization by local directional refinement. T3 - ZIB-Report - SC-96-16 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2270 ER - TY - CHAP A1 - Fröhlich, Jochen A1 - Deuflhard, Peter ED - Neunzert, H. T1 - An adaptive Galerkin method for the PDF transport equation of turbulent reactive flows T2 - Progress in Industrial Mathematics, Proc. 8th Conference of the European Consortium for Mathematics in Industry (ECMI 94) Y1 - 1996 SP - 176 EP - 184 PB - Wiley & Teubner ER -