Philipp Birken | Lund University (original) (raw)

Papers by Philipp Birken

PAMM, 2007

We consider the solution of a sequence of nonsymmetric linear systems arising from a supersonic m... more We consider the solution of a sequence of nonsymmetric linear systems arising from a supersonic model problem by exploiting triangular preconditioner updates. In addition, we demonstrate how the power of the updates can be enhanced by permuting the entire sequence beforehand with a physically motivated reordering of unknowns.

Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2013

ABSTRACT The standard Godunov type method used in computational fluid dynamics shows accuracy pro... more ABSTRACT The standard Godunov type method used in computational fluid dynamics shows accuracy problems for low Mach number flows and for the kinetic energy at the highest wave numbers resolvable on a given grid. Both drawbacks become visible when simulating the decay of isotropic turbulence at the low Mach numbers typical for the respective experimental investigations. A modification to cure both problems is proposed by Thornber et al. [10] with a mathematical motivation in case of a special fifth order reconstruction. The theoretical results are repeated here. Numerical results are achieved for schemes not investigated in that literature, namely AUSMDV and AUSM+-up which includes already modifications for low Mach number flows. First experiences with Thornber’s modification confirm the positive influence in combination with AUSMDV even if the reconstruction is only of second order. In combination with AUSM+-up Thornber’s modification provides too little damping.

Proceedings of Symposia in Applied Mathematics, 2009

PAMM, 2009

We explore the idea of using nonlinear schemes as preconditioners in Newton-Krylov schemes for un... more We explore the idea of using nonlinear schemes as preconditioners in Newton-Krylov schemes for unsteady flow computations. Analysis shows that left preconditioning changes the Newton scheme in a non equivalent way, leading to a stall in Newton convergence, whereas right preconditioning leads to a sound method.

Numerical Methods for Partial Differential Equations, 2014

We present a parallel matrix-free implicit finite volume scheme for the solution of unsteady thre... more We present a parallel matrix-free implicit finite volume scheme for the solution of unsteady three-dimensional advection-diffusion-reaction equations with smooth and Dirac-Delta source terms. The scheme is formally second order in space and a Newton-Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix-vector product required is hardcoded without any approximations, obtaining a matrix-free method that needs little storage and is well suited for parallel implementation. We describe the matrix-free implementation of the method in detail and give numerical evidence of its second order convergence in the presence of smooth source terms. For non-smooth source terms the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long time simulation of calcium flow in heart cells and show its parallel scaling.

PAMM, 2012

We consider the use of DIRK and Rosenbrock schemes for the time integration of unsteady flow prob... more We consider the use of DIRK and Rosenbrock schemes for the time integration of unsteady flow problems. Thereby, two variants of solving the linear systems are compared regarding their efficiency, namely a Jacobian-free method versus computing an approximate Jacobian. The Rosenbrock schemes are slow in the Jacobian-free variant, but may be competitive in the other.

The thermal coupling of a fluid and a structure is of great significance for many industrial proc... more The thermal coupling of a fluid and a structure is of great significance for many industrial processes. As a model for cooling processes in heat treatment of steel we consider the surface coupling of the compressible Navier-Stokes equations bordering at one part of the surface with the heat equation in a solid region. The semi-discrete coupled system is solved using stiffly stable SDIRK methods of higher order, where on each stage a fluid-structurecoupling problem is solved. For the resulting method it is shown by numerical experiments that a second order convergence rate is obtained. This property is further used to implement a simple time-step control, which saves considerable computational time and, at the same time, guarantees a specified maximum error of time integration.

PAMM, 2005

ABSTRACT A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The m... more ABSTRACT A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments, as well as by analysis, that the preconditioned scheme yields a physically corrected pressure distribution and combined with an explicit time integrator it is stable if the time step Δt satisfies the requirement to be (M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Δt = (M ),M → 0, though producing unphysical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This contribution compares the efficiency of Rosenbrock time integration schemes with ESDIRK sche... more This contribution compares the efficiency of Rosenbrock time integration schemes with ESDIRK schemes, applicable to unsteady flow and fluid-structure interaction simulations. Compared to non-linear ESDIRK schemes, the linear implicit Rosenbrock-Wanner schemes require subsequent solution of the same linear systems with different right hand sides. By solving the linear systems with the iterative solver GMRES, the preconditioner can be reused for the subsequent stages of the Rosenbrock-Wanner scheme. Unsteady flow simulations show a gain in computational efficiency of approximately factor three to five in comparison with ESDIRK.

We consider time dependent thermal fluid structure interaction. The respective models are the com... more We consider time dependent thermal fluid structure interaction. The respective models are the compressible Navier-Stokes equations and the nonlinear heat equation. A partitioned coupling approach via a Dirichlet-Neumann method and a fixed point iteration is employed. As a refence solver a previously developed efficient time adaptive higher order time integration scheme is used.

A modification of the Roe scheme aimed at low Mach number flows is discussed. It improves the dis... more A modification of the Roe scheme aimed at low Mach number flows is discussed. It improves the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. No conflict is observed between the reduced dissipation and the accuracy or stability of the scheme in any of the investigated test cases ranging from low Mach number potential flow to hypersonic viscous flow around a cylinder.

ABSTRACT We consider the coupling of the compressible Navier‐Stokes equations with the heat equat... more ABSTRACT We consider the coupling of the compressible Navier‐Stokes equations with the heat equation as a model for cooling processes in steel forming. The semidiscrete coupled system is solved using stiffly stable SDIRK methods of higher order, where on each stage, a fluid‐structure interaction problem is solved.

In this article, the coupling of the temperature-dependent, compressible Navier-Stokes equations ... more In this article, the coupling of the temperature-dependent, compressible Navier-Stokes equations solved by a compressible finite volume scheme together with the finite element solution of the heat equation is considered. The application is focused on the cooling process of a heated metal bar treated in the field of metal forming technology. This is done both by loose and strong numerical coupling methods based on the Backward-Euler scheme, where, particularly, Gauss-Seidel and fixed-point solvers are considered.

PAMM, 2010

International Journal for Numerical Methods in Fluids, 2009

The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Ne... more The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Newton-Krylov-solvers is considered and analyzed. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix-free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes which are demonstrated through numerical results.

BIT Numerical Mathematics, 2005

, Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as wel... more , Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step ∆t does not satisfy the requirement to be O(M 2 ) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to ∆t = O(M ), M → 0, which results from the well-known CFL-condition.

Applied Numerical Mathematics, 2008

This paper deals with solving sequences of nonsymmetric linear systems with a block structure ari... more This paper deals with solving sequences of nonsymmetric linear systems with a block structure arising from compressible flow problems. The systems are solved by a preconditioned iterative method. We attempt to improve the overall solution process by sharing a part of the computational effort throughout the sequence. Our approach is fully algebraic and it is based on updating preconditioners by a block triangular update. A particular update is computed in a black-box fashion from the known preconditioner of some of the previous matrices, and from the difference of involved matrices. Results of our test compressible flow problems show, that the strategy speeds up the entire computation. The acceleration is particularly important in phases of instationary behavior where we saved about half of the computational time in the supersonic and moderate Mach number cases. In the low Mach number case the updated decompositions were similarly effective as the frozen preconditioners.

Proceedings of symposia in applied mathematics, Nov 19, 2009

List of all HYP2008 Participants Name Aﬃliation Status Debora Amadori Fabio Ancona Stuart Antman ... more List of all HYP2008 Participants Name Aﬃliation Status Debora Amadori Fabio Ancona Stuart Antman Paolo Antonelli Paul Arminjon Agissilaos Athanassoulis Prashant Athavale Jorge Balbas Weizhu Bao Sylvie Benzoni-Gavage Stefan Berres Philipp Birken Animikh Biswas Andreas Bollermann Benjamin Boutin Jerome Breil Alberto Bressan Raimund Burger Miroslav Cada Tony Chan Grigori Chapiro Li Chen Gui-Qiang Chen Shuxing Chen Jing Chen Juan Cheng Bin Cheng Alina Chertock Kyu Yong Choi Cleopatra Christoforou Rinaldo Colombo Jeﬀery Cooper ...