Margreet Nool

Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation

Abstract Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5–8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the ∇· B =0 constraint are examined. Three source term approaches suitable for cell-centered B representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax–Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.

A Parallel Jacobi--Davidson-type Method for Solving Large Generalized Eigenvalue Problems in Magnetohydrodynamics

We study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form

Calculation of resistive magnetohydrodynamic spectra in tokamaks

A x = \lambda B x

Parallel Jacobi-Davidson for Solving Generalized Eigenvalue Problems

, in which the complex matrices A and B are block-tridiagonal, and B is Hermitian positive definite. The Jacobi--Davidson method appears to be an excellent method for parallel computation of a few selected eigenvalues because the basic ingredients are matrix vector products, vector updates, and inner products. The method is based on solving projected eigenproblems of order typically less than 30. We apply a complete block LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable algorithm. One decomposition can be used for the calculation of several eigenvalues. Spectral transformations are presented to compute certain interior eigenvalues and their associated eigenvectors. The convergence behavior of several variants of the Jacobi--Davidson algorithm is examined. Special attention is paid to the parallel performance, memory requirements, and prediction of the speed-up. Numerical results obtained on a distributed memory Cray T3E are shown.

Parallel Implementation of a Least-Squares Spectral Element Solver for Incompressible Flow Problems

Resistive magnetohydrodynamic spectra of toroidal plasmas are calculated using the recently developed Jacobi–Davidson eigenvalue solver. Poloidal mode coupling in finite aspect ratio tokamaks yields gaps in the ideal Alfven continuous spectrum. If resistivity is included, the ideal continua disappear and are replaced by damped global waves located on specific curves in the complex frequency plane. The end points of these curves join the tips of the ideal continua and the boundaries of the ideal spectral gap. The eigenfunctions of the waves on these resistive curves are shown to have definite parity in the poloidal harmonics. It is shown that for very small toroidicity the topology of the resistive spectrum is completely different from the cylindrical one. Independent of the size of the inverse aspect ratio the ideal gap remains visible in the resistive spectrum.

AMRVAC: A multidimensional grid-adaptive magnetofluid dynamics code

We study the Jacobi-Davidson method for the solution of large generalised eigenproblems as they arise in MagnetoHydroDynamics. We have combined Jacobi-Davidson (using standard Ritz values) with a shift and invert technique. We apply a complete LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelisable algorithm. Moreover, we describe a variant of Jacobi-Davidson in which harmonic Ritz values are used. In this variant the same parallel LU decomposition is used, but this time as a preconditioner to solve the ‘correction‘ equation.

Flecs, a flexible coupling shell application to fluid-structure interaction

Least-squares spectral element methods (LSQSEM) are based on two important and successful numerical methods: spectral/hp element methods and least-squares finite element methods. Least-squares methods lead to symmetric and positive definite algebraic systems which circumvent the Ladyzhenskaya–Babuška–Brezzi (LBB) stability condition and consequently allow the use of equal order interpolation polynomials for all variables. In this paper, we present results obtained with a parallel implementation of the least-squares spectral element solver on a distributed memory machine (Cray T3E) and on a virtual shared memory machine (SGI Origin 3800).

A parallel least-squares spectral element solver for incompressible flow problems on unstructured grids

Abstract We present the results obtained with AMRVAC, a software package designed for solution-adaptive time-accurate (magneto)hydrodynamic simulations. In any dimensionality, the grid adjusts to capture shocks and other sharp flow features accurately following an automated Adaptive Mesh Refinement [AMR] strategy. This grid adaptation algorithm is incorporated with the Versatile Advection Code [VAC], so that it can be used to time-advance sets of conservation laws with options for the spatial discretization employed. We demonstrate and evaluate the efficiency achievable by AMR for 1D, 2D, and 3D test problems and describe the employed data structures.

A parallel, state-of-the-art, least-squares spectral element solver for incompressible flow problems

Numerical simulations involving multiple, physically different domains can be solved effectively by coupling simulation programs, or solvers. The coordination of the different solvers is commonly handled by a coupling shell. A coupling shell synchronizes the execution of the solvers and handles the transfer of data from one physical domain to another. In this paper, we introduce Flecs, a flexible coupling shell, designed for implementing and applying an interface for multidisciplinary simulations with superior accuracy. The aim is not to achieve the best possible efficiency or to support a large feature set, but to provide a flexible platform for developing new data transfer algorithms and coupling schemes.

PPPS-2013: Particle and hybrid modeling of NS pulsed discharges: Discharge structures and electron energies

The parallelisation of the least-squares spectral element formulation of the Stokes problem is discussed for incompressible flow problems on unstructured grids. The method leads to a large symmetric positive definite algebraic system, that is solved iteratively by the conjugate gradient method. To improve the convergence rate, both Jacobi and Additive Schwarz preconditioners are applied. Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large-scale incompressible flows on a Cray T3E and on an SGI Origin 3800.