| 1. |
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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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| 11. |
- Birken, Philipp, et al.
(författare)
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Preconditioned Smoothers for the Full Approximation Scheme for the RANS Equations
- 2019
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 78:2, s. 995-1022
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Tidskriftsartikel (refereegranskat)abstract
- We consider multigrid methods for finite volume discretizations of the Reynolds averaged Navier–Stokes equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge–Kutta (AERK) methods. Furthermore, by deriving them from Rosenbrock smoothers, we identify some existing schemes as a class of additive W (AW) methods. This gives rise to two classes of preconditioned smoothers, preconditioned AERK and AW, which are implemented the exact same way, but have different parameters and properties. This derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. Numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations.
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| 12. |
- Birken, Philipp, et al.
(författare)
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Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods
- 2019
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Ingår i: International Journal of Computational Fluid Dynamics. - : Informa UK Limited. - 1061-8562 .- 1029-0257. ; 33:9, s. 353-361
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Tidskriftsartikel (refereegranskat)abstract
- We suggest a new multigrid preconditioning strategy for use in Jacobian-free Newton–Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretisations. To define the new preconditioner, use is made of an auxiliary first-order finite volume discretisation that refines the original DG mesh, but can still be implemented algebraically. As smoother, we consider the pseudo-time iteration W3 with a symmetric Gauss–Seidel-type approximation of the Jacobian. As a proof of concept numerical tests are presented for the one-dimensional Euler equations, demonstrating the potential of the new approach.
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| 13. |
- Birken, Philipp
(författare)
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Termination criteria for inexact fixed-point schemes
- 2015
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Ingår i: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325. ; 22:4, s. 702-716
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Tidskriftsartikel (refereegranskat)abstract
- We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x=F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion.
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| 14. |
- Blom, David S., et al.
(författare)
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A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows
- 2016
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Ingår i: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1572-9044 .- 1019-7168. ; 42:6, s. 1401-1426
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.
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| 15. |
- Frenander, Hannes
(författare)
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High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations.For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT).A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time.To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time.In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.
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| 16. |
- Gleim, Tobias, et al.
(författare)
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Experimental and Numerical Aspects of a Thermal Fluid-Structure Phenomenon
- 2017
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Ingår i: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2016). - : Author(s). - 0094-243X .- 1551-7616. - 9780735415386 ; 1863
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Konferensbidrag (refereegranskat)abstract
- A fundamental research experiment for thermal fluid-structure-interaction for the verification of a partitioned approach with non-linear material properties is examined. In the following, a specimen is heated as well as cooled within a wind tunnel. The thermal fluid-structure-interaction is first experimentally investigated and subsequently numerically validated. For the numerical simulation, two existing programs (a fluid and a structure code) are coupled using a partitioned approach.
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| 17. |
- Gleim, Tobias, et al.
(författare)
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Thermal Fluid-Structure-Interaction - Experimental and Numerical Analysis
- 2015
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Ingår i: Proceedings of the International Conference of Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014). - : AIP Publishing LLC. - 0094-243X .- 1551-7616. ; 1648
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Konferensbidrag (refereegranskat)abstract
- In the present paper the thermal fluid-structure-interaction is experimentally and numerically investigated. Therefore, the interaction phenomena is modeled by the Reynolds-averaged Navier-Stokes equations and the nonlinear Fourier heat conduction equation are used for the fluid and the solid phase, respectively. The simulation is performed using a partitioned approach using the finite volume method for the fluid domain, the finite element method for the solid domain and Runge-Kutta integration schemes for the time domain. Furthermore, as a basis for the understanding of thermal fluid-structure-interaction and also for the veri cation and validation of the applied continuum mechanical models and numerical methods, respectively, a fundamental wind tunnel experiment is presented.
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| 18. |
- Görtz, Morgan, et al.
(författare)
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On the convergence rate of the dirichlet-neumann iteration for coupled poisson problems on unstructured grids
- 2020
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Ingår i: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, FVCA 2020. - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783030436506 ; 323, s. 355-363
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Konferensbidrag (refereegranskat)abstract
- We consider thermal fluid structure interaction with a partitioned approach, where typically, a finite volume and a finite element code would be coupled. As a model problem, we consider two coupled Poisson problems with heat conductivities $$\lambda _1$$, $$\lambda _2$$ in one dimension on intervals of length $$l:1$$ and $$l:2$$. Hereby, we consider linear discretizations on arbitrary meshes, such as finite volumes, finite differences, finite elements. For these, we prove that the convergence rate of the Dirichlet-Neumann iteration is given by $$\lambda _1l_2/\lambda _2l_1$$ and is thus independent of discretization and mesh.
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| 19. |
- Kasimir, Johannes, et al.
(författare)
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An finite volume based multigrid preconditioner for dg-sem for convection-diffusion
- 2021
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Ingår i: Fluid Dynamics and Transport Phenomena. - : CIMNE. ; 600, s. 1-12
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Konferensbidrag (refereegranskat)abstract
- The goal of our research is the construction of efficient Jacobian-free preconditioners for high order Discontinuous Galerkin (DG) discretizations with implicit time integration. We are motivated by three-dimensional unsteady compressible flow applications, which often result in large stiff systems. Implicit time integrators overcome the impact upon restrictive CFL conditions on explicit ones but leave the problem to solve huge nonlinear systems. In this paper we consider a multigrid preconditioning strategy for Jacobian-free Newton-Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretizations. The preconditioner is defined by an auxiliary first order Finite Volume (FV) discretization that refines the original DG mesh, but can still be implemented algebraically. Different options exist to define the grid transfer between DG and FV. We suggest an ad hoc assignment of the unknowns as well as L2 projections. We present new numerical results for the two-dimensional convection-diffusion equation in combination with the different transfer options, which demonstrate the quality and efficiency of the suggested preconditioner with regards to convergence speed up and CPU time. The suggested L2 projection from this paper result in the best convergence speed up.
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| 20. |
- Linders, Viktor, et al.
(författare)
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Locally conservative and flux consistent iterative methods
- 2023
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow.
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| 21. |
- Linders, Viktor, et al.
(författare)
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LOCALLY CONSERVATIVE AND FLUX CONSISTENT ITERATIVE METHODS
- 2024
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275. ; 46:2, s. 424-444
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Tidskriftsartikel (refereegranskat)abstract
- Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton-Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow.
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| 22. |
- Linders, Viktor, et al.
(författare)
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On the Consistency of Arnoldi-Based Krylov Methods for Conservation Laws
- 2023
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Ingår i: PAMM - Proceedings in Applied Mathematics and Mechanics. - 1617-7061. ; 23:1
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Tidskriftsartikel (refereegranskat)abstract
- Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Re- cently, these concepts have been extended to the realm of iterative methods by defining locally conservative and flux consistent iterations. In this note, the current status of such iterative methods is summarized. In particular, it has been shown that Krylov subspace methods are locally conservative, but that they are not flux consistent. Here, we approach the problem of quantifying the flux inconsistency of Krylov subspace methods. Krylov methods introduce a time retardation factor into discretizations of linear conservation laws. It has thusfar been unknown how to compute the precise value of this factor. This issue is resolved herein for Arnoldi-based Krylov subspace methods.
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| 23. |
- Linders, Viktor, et al.
(författare)
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Resolving entropy growth from iterative methods
- 2023
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Ingår i: BIT Numerical Mathematics. - 0006-3835. ; 63:4
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Tidskriftsartikel (refereegranskat)abstract
- We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.
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| 24. |
- Linders, Viktor, et al.
(författare)
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Resolving Entropy Growth from Iterative Methods
- 2023
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.
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| 25. |
- Meisrimel, Peter, et al.
(författare)
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A time adaptive multirate Dirichlet–Neumann waveform relaxation method for heterogeneous coupled heat equations
- 2023
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Ingår i: ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik. - 0044-2267. ; 103:11
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Tidskriftsartikel (refereegranskat)abstract
- We consider partitioned time integration for heterogeneous coupled heat equations. First and second order multirate, as well as time-adaptive Dirichlet-Neumann Waveform relaxation (DNWR) methods are derived. In 1D and for implicit Euler time integration, we analytically determine optimal relaxation parameters for the fully discrete scheme. We test the robustness of the relaxation parameters on the second order multirate method in 2D. DNWR is shown to be very robust and consistently yielding fast convergence rates, whereas the closely related Neumann-Neumann Waveform relaxtion (NNWR) method is slower or even diverges. The waveform approach naturally allows for different timesteps in the subproblems. In a performance comparison for DNWR, the time-adaptive method dominates the multirate method due to automatically finding suitable stepsize ratios. Overall, we obtain a fast, robust, multirate and time adaptive partitioned solver for unsteady conjugate heat transfer.
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