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| 2. |
- Birken, Philipp, et al.
(författare)
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A Note on Terminology in Multigrid Methods
- 2016
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Ingår i: PAMM - Proceedings in Applied Mathematics and Mechanics. - : Wiley. - 1617-7061. ; 16, s. 721-722
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Tidskriftsartikel (refereegranskat)abstract
- We compare terminology used in the literature on multigrid methods for compressible computational fluid dynamics to that used in linear multigrid theory. Several popular iterative and direct smoothers are presented side-by-side using the same terminology. We argue for greater analysis of these methods in order to place them into a more rigorous framework and to identify the most promising candidates for future development.
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| 3. |
- Birken, Philipp, et al.
(författare)
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A STUDY of multigrid smoothers used in compressible CFD based on the convection diffusion equation
- 2016
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Ingår i: ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering. - 9786188284401 ; 2, s. 2648-2663
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Konferensbidrag (refereegranskat)abstract
- We look at multigrid methods for unsteady viscous compressible flows. We specifically target smoothers that can be used in parallel and without computation of a Jacobian, which are particlarly attractive candidates in the context of Discontinuous Galerkin discretizations. In CFD, a plethora of nonlinear smoothers have been suggested which are hard to analyze. Our methodology is to use a linear model problem, here the convection diffusion equation, to be able to classify and compare smoothers better. Specifically, we consider explicit and implicit pseudo time iterations, GMRES as a smoother, SGS and implicit line smoothers. We relate GMRES to explicit Runge-Kutta smoothers, identify implicit line smoothers as Block Jacobi and analyze the potential of implicit pseudo time iterations. Finally, we discuss the relation between methods for steady and unsteady flows. Numerical results show that GMRES is a very attractive smoother in this context.
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| 4. |
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| 5. |
- Birken, Philipp, et al.
(författare)
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Conservation Properties of Iterative Methods for Implicit Discretizations of Conservation Laws
- 2022
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 92:2
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Tidskriftsartikel (refereegranskat)abstract
- Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton’s method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the corresponding numerical flux can be inconsistent with the conservation law. We prove an extension of the Lax-Wendroff theorem, which reveals that numerical solutions based on these methods converge to weak solutions of a modified conservation law where the flux function is multiplied by a particular constant. This constant depends on the choice of Runge-Kutta method but is independent of both the conservation law and the discretization. Consistency is maintained by ensuring that this constant equals unity and a strategy for achieving this is presented. Simulations show that this strategy improves the convergence rate of the pseudo-time iterations. Experiments with GMRES suggest that it also suffers from inconsistency but that this is automatically accounted for after some finite number of iterations. Similar experiments with coarse grid corrections based on agglomeration indicate no inconsistency.
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| 6. |
- Birken, Philipp, et al.
(författare)
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Extrapolation in Time in Thermal Fluid Structure Interaction
- 2015
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Ingår i: Recent Trends in Computational Engineering - CE2014 : Optimization, Uncertainty, Parallel Algorithms, Coupled and Complex Problems (Lecture Notes in Computational Science and Engineering). - Cham : Springer International Publishing. ; 105, s. 215-231
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Konferensbidrag (refereegranskat)abstract
- 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 reference solver a previously developed efficient time adaptive higher order time integration scheme is used. To improve upon this, we work on reducing the number of fixed point coupling iterations. Thus, we explore the idea of extrapolation based on data given from the time integration and derive such methods for SDIRK2. This allows to reduce the number of fixed point iterations further by up to a factor of two with linear extrapolation performing better than quadratic.
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| 7. |
- Birken, Philipp, et al.
(författare)
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Fast Solvers for Unsteady Thermal Fluid Structure Interaction
- 2015
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Ingår i: International Journal for Numerical Methods in Fluids. - : Wiley. - 1097-0363 .- 0271-2091. ; 79:1, s. 16-29
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Tidskriftsartikel (refereegranskat)abstract
- 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 reference solver, a previously developed efficient time-adaptive higher-order time integration scheme is used. To improve on this, we work on reducing the number of fixed point coupling iterations. Using the idea of extrapolation based on data given from the time integration by deriving such methods for SDIRK2, it is possible to reduce the number of fixed point iterations further by up to a factor of two with linear extrapolation performing better than quadratic. This leads to schemes that can use less than two iterations per time step. Furthermore, widely used vector extrapolation methods for convergence acceleration of the fixed point iteration are tested, namely Aitken relaxation, minimal polynomial extrapolation and reduced rank extrapolation. These have no beneficial effects.
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| 8. |
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| 9. |
- Birken, Philipp, et al.
(författare)
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Numerical methods for unsteady thermal fluid structure interaction
- 2017
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Ingår i: Fluid-Structure Interaction Modeling, Adaptive Discretisations and Solvers. - : De Gruyter. - 9783110494259 ; , s. 129-168
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Bokkapitel (refereegranskat)abstract
- We discuss thermal fluid-structure interaction processes, where a simulation of the time-dependent temperature field is of interest. Thereby, we consider partitioned coupling schemes with a Dirichlet-Neumann method. We present an analysis of the method on a model problem of discretized coupled linear heat equations. This shows that for large quotients in the heat conductivities, the convergence rate will be very small. The time dependencymakes the use of time-adaptive implicitmethods imperative. This gives rise to the question as to how accurately the appearing nonlinear systems should be solved, which is discussed in detail for both the nonlinear and linear case. The efficiency of the resulting method is demonstrated using realistic test cases.
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| 10. |
- Birken, Philipp, et al.
(författare)
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On stability and conservation properties of (S)epirk integrators in the context of discretized pdes
- 2018
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Ingår i: Theory, Numerics and Applications of Hyperbolic Problems II. - Cham : Springer International Publishing. - 9783319915470 ; 237, s. 617-629
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Konferensbidrag (refereegranskat)abstract
- Exponential integrators are becoming increasingly popular for stiff problems of high dimension due to their attractive property of solving the linear part of the system exactly and hence being A-stable. In practice, however, exponential integrators are implemented using approximation techniques to matrix-vector products involving functions of the matrix exponential (the so-called ϕ-functions) to make them efficient and competitive to other state-of-the-art schemes. We will examine linear stability and provide a Courant–Friedrichs–Lewy (CFL) condition of special classes of exponential integrator schemes called EPIRK and sEPIRK and demonstrate their dependence on the parameters of the embedded approximation technique. Furthermore, a conservation property of the EPIRK schemes is proven.
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