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In this paper we present the main conceptual ingredients and the current state of development of the new solver QUADFLOW for large scale simulations of compressible fluid flow and fluid--structure interaction. In order to keep the size of the discrete problems at every stage as small as possible for any given target accuracy, we employ a multiresolution adaptation strategy that will be described in the first part of the paper. In the second part we outline a new mesh generation concept that is to support the adaptive concepts as well as possible. A key idea is to understand meshes as parametric mappings determined by possibly few control points as opposed to store each mesh cell separately. Finally, we present a finite volume discretization along with suitable data structures which again is to support the adaptation concepts. We conclude with numerical examples of realistic applications demonstrating different features of the solver.

We are interested in solving the Vlasov equation used to describe collective effects in plasmas. This nonlinear partial differential equation coupled with Maxwell equation describes the time evolution of the particle distribution in phase space. The numerical solution of the full three-dimensional Vlasov-Maxwell system represents a considerable challenge due to the huge size of the problem. A numerical method based on wavelet transform enables to compute the distribution function on an adaptive mesh from a regular discretization of the phase space. In this paper, we evaluate the costs of this recently developed adaptive scheme applied on a reduced one-dimensional model, and its parallelization. We got a fine grain parallel application that achieves a good scalability up to 64 processors on a shared memory architecture. 1

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An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two-dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L ∞ metric. The numerical solutions are proved to converge in L ∞ towards the exact ones as ε and ∆t tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to W 1, ∞ ∩W 2,1. The rate of convergence is O(∆t 2 + ε/∆t), which should be compared to the results of Besse, who recently established [6] similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in

Abstract. The concept of fully adaptive multiscale finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In order to perform 3D computations for complex geometries in reasonable CPU time, the underlying multiscale-based grid adaptation strategy has to be parallelised via MPI for distributed memory architectures. In view of a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be well-balanced and communication between processors has to be minimised. This has been realised using space-filling curves.

In this paper we continue our work on adaptive timestep control for weakly instationary problems [29]. The core of the method is a space-time splitting of adjoint error representations for target functionals due to Süli [31] and Hartmann [18]. The main new ingredients are (i) the extension from scalar, 1D, conservation laws to the 2D Euler equations of gas dynamics, (ii) the derivation of boundary conditions for a new formulation of the adjoint problem and (iii) the coupling of the adaptive time-stepping with spatial adaptation. For the spatial adaptation, we use a multiscale-based strategy developed by Müller [24], and we combine this with an implicit time discretization. The combined space-time adaptive method provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and becomes small when a perturbation enters the flow field. The efficiency of the solver is investigated by means of an unsteady inviscid 2D flow over a bump.

Summary. In the context of offshore oil production, we are interested in accurate and fast computation of two-phase flows in pipelines. A one dimensional model of hyperbolic equations is solved numerically by an explicit Lagrange- Euler projection method. This paper shows that adaptive multiresolution techniques can speed up the computation significantly. Even more so when local time stepping enhancement is used. 1 Modeling of the physical problem In this short paper we restrict ourselves to a homogeneous model for two-phase flows. The density ρ, velocity u and the gas mass fraction Y of the mixture of oil and gas are related through a PDE system ⎨∂t(ρ)

1 Introduction The method to be analyzed here was introduced by Mats Holmstr"om in his PhD thesis [9], and it was called the SPR method, for sparse point representation. The SPR method is an adaptive finite difference strategy for the numerical solution of evolution partial differential equations ut(x; t) = Lu(x; t); t? 0; x 2 \Omega (1) augmented with initial and boundary conditions. The method combines the simplicity and accuracy of traditional finite difference schemes with the ability of wavelet coefficients in the characterization of local regularity of functions. In Holmstr"om's case, the differential equation is of hyperbolic type, and the method was used to simulate some typical models in fluid dynamics, showing savings in CPU time when compared with the standard finite difference scheme.

Abstract The concept of the new fully adaptive flow solver Quadflow has been developed within the SFB 401 over the past 12 years. Its primary novelty lies in the integration of new and advanced mathematical tools in a unified environment. This means that the core ingredients of the finite volume solver, the grid adaptation and grid generation are adapted to each others needs rather than putting them together as independent black boxes. In this paper we shall present recent developments and demonstrate their efficiency by numerical experiments for some representative basic configurations. 1