8 research outputs found
Non-Modal Analysis of Multigrid Schemes for the High-Order Flux Reconstruction Method
The present study introduces an application of the non-modal analysis to multigrid operators with explicit Runge-Kutta smoothers in the context of Flux Reconstruction discretizations of the linear convection-diffusion equation. A dissipation curve is obtained that reflects upon the convergence properties of the multigrid operator. The number of smoothing steps, the type of cycle (V/W) and the combination of pand h-multigrid are taken into account in order to find those configurations which yield faster convergence rates. The analysis is carried out for polynomial orders up to P = 6, in 1D and 2D for varying degrees of convection (Péclet number), as well as for high aspect ratio cells. The non-modal analysis can support existing evidence on the behaviour of multigrid schemes. W-cycles, a higher number of coarse-level sweeps or the combined use of hp-multigrid are shown to increase the error dissipation, while higher degrees of convection and/or high aspect-ratio cells both decrease the error dissipation rate
Explicit-Implicit Domain Splitting for Two Phase Flows with Phase Transition
Two phase flows that include phase transition, especially phase creation,
with a sharp interface remain a challenging task for numerics. We consider the
isothermal Euler equations with phase transition between a liquid and a vapor
phase. The phase interface is modeled as a sharp interface and the mass
transfer across the phase boundary is modeled by a kinetic relation. Existence
and uniqueness results were proven in Ref. \cite{Hantke2019a}. Using sharp
interfaces for simulating nucleation and cavitation results in the grid
containing tiny cells that are several orders of magnitude smaller than the
remaining grid cells. This forces explicit time stepping schemes to take tiny
time steps on these cells. As a remedy we suggest an explicit implicit domain
splitting where the majority of the grid cells is treated explicitly and only
the neighborhood of the tiny cells is treated implicitly. We use dual time
stepping to solve the resulting small implicit systems. Our numerical results
indicate that the new scheme is robust and provides significant speed-up
compared to a fully explicit treatment
An implicit HDG method for linear convection-diffusion with dual time stepping
This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a von Neumann analysis, leading to a closed form expression for the critical dual time step. An optimal choice for the dual time step, producing the maximum damping for all the frequencies, is also derived. Steady and transient convection-diffusion problems are considered to demonstrate the performance of the proposed DTS approach, with particular emphasis on convection dominated problems. Two simple approaches to accelerate the convergence of the DTS approach are also considered and three different time marching approaches for the dual time are compared
Artificial Compressibility Approaches in Flux Reconstruction for Incompressible Viscous Flow Simulations
Copyright © 2021 The Author(s). Several competing artificial compressibility methods for the incompressible flow equations are examined using the high-order flux reconstruction method. The established artificial compressibility method (ACM) of \citet{Chorin1967} is compared to the alternative entropically damped (EDAC) method of \citet{Clausen2013}, as well as an ACM formulation with hyperbolised diffusion. While the former requires the solution to be converged to a divergence free state at each physical time step through pseudo iterations, the latter can be applied explicitly. We examine the sensitivity of both methods to the parameterisation for a series of test cases over a range of Reynolds numbers. As the compressibility is reduced, EDAC is found to give linear improvements in divergence whereas ACM yields diminishing returns. For the Taylor--Green vortex, EDAC is found to perform well; however on the more challenging circular cylinder at , EDAC gives rise to early transition of the free shear-layer and over-production of the turbulence kinetic energy. This is attributed to the spatial pressure fluctuations of the method. Similar behaviour is observed for an aerofoil at with an attached transitional boundary layer. It is concluded that hyperbolic diffusion of ACM can be beneficial but at the cost of case setup time, and EDAC can be an efficient method for incompressible flow. However, care must be taken as pressure fluctuations can have a significant impact on physics and the remedy causes the governing equation to become overly stiff.https://arxiv.org/abs/2111.07915v
Locally adaptive pseudo-time stepping for high-order Flux Reconstruction
This paper proposes a novel locally adaptive pseudo-time stepping convergence acceleration technique for dual time stepping which is a common integration method for solving unsteady low-Mach preconditioned/incompressible Navier-Stokes formulations. In contrast to standard local pseudo-time stepping techniques that are based on computing the local pseudo-time steps directly from estimates of the local Courant-Friedrichs-Lewy limit, the proposed technique controls the local pseudo-time steps using local truncation errors which are computed with embedded pair RK schemes. The approach has three advantages. First, it does not require an expression for the characteristic element size, which are difficult to obtain reliably for curved mixedelement meshes. Second, it allows a finer level of locality for high-order nodal discretisations, such as FR, since the local time-steps can vary between solution points and field variables. Third, it is well-suited to being combined with P-multigrid convergence acceleration. Results are presented for a laminar 2D cylinder test case at Re = 100. A speed-up factor of 4.16 is achieved compared to global pseudo-time stepping with an RK4 scheme, while maintaining accuracy. When combined with P-multigrid convergence acceleration a speed-up factor of over 15 is achieved. Detailed analysis of the results reveals that pseudo-time steps adapt to element size/shape, solution state, and solution point location within each element. Finally, results are presented for a turbulent 3D SD7003 airfoil test case at Re = 60, 000. Speed-ups of similar magnitude are observed, and the flow physics is found to be in good agreement with previous studies
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Numerical Analysis of Flux Reconstruction
High-order methods have become of increasing interest in recent years in computational
physics. This is in part due to their perceived ability to, in some cases, reduce the computational overhead of complex problems through both an efficient use of computational
resources and a reduction in the required degrees of freedom. One such high-order
method in particular – Flux Reconstruction – is the focus of this thesis. This body of work
relies and expands on the theoretical methods that are used to understand the behaviour
of numerical methods – particularly related to their real-world application to industrial
problems.
The thesis begins by challenging some of the existing dogma surrounding computational fluid dynamics by evaluating the performance of high-order flux reconstruction.
First, the use of the primitive variables as an intermediary step in the construction of flux
terms is investigated. It is found that reducing the order of the flux function by using the
conserved rather than primitive variables has a substantial impact on the resolution of
the method. Critically, this is supported by a theoretical analysis, which shows that this
mechanism of error generation becomes increasing important to consider as the order of
accuracy increases.
Next, the analysis of Flux Reconstruction was extended by analytically and numerically exploring the impact of higher dimensionality and grid deformation. It is found
that both expanding and contracting grids – essential components of real-world domain
decomposition – can cause dispersion overshoot in two dimensions, but that FR appears
to suffer less that comparable Finite Difference approaches. Fully discrete analysis is then
used to show that, depending on the correction function, small perturbations in incidence
angle can cause large changes in group velocity. The same analysis is also used to theoretically demonstrate that Discontinuous Galerkin suffers less from dispersion error than
Huynh’s FR scheme – a phenomenon that has previously been observed experimentally,
but not explained theoretically.
This thesis concludes with the presentation of a robust theoretical underpinning for
determining stable correction functions for FR. Three new families of correction functions
are presented, and their properties extensively explored. An important theoretical finding
is introduced – that stable correction functions are not defined uniquely be a norm. As a
result, a generalised approach is presented, which is able to recover all previously defined
correction functions, but in some instances via a different norm to their original derivation.
This new super-family of correction functions shows considerable promise in increasing
temporal stability limits, reducing dispersion when fully discretised, and increasing the
rate of convergence.
Taken altogether, this thesis represents a considerable advance in the theoretical
characterisation and understanding of a numerical method – one which, it has been shown,
has enormous potential for forming the heart of future computational physics codes