International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 8th Dec 2022, 11:44:26pm CET
MS 1a: stable and efficient time integration schemes for conservation laws and related models
Ever since the seminal work of Tadmor  there has been some interest in entropy conserva-
tion/dissipation of semidiscretisations for hyperbolic conservation laws and related models.
Some recent contributions to this active research topic include methods based on summation
by parts (SBP) operators in space and residual distribution (RD) schemes, both forming very
general frameworks of semidiscrete numerical methods.
However, there are less results concerning fully discrete entropy conservation/stability.
While there are some estimates for schemes containing a considerable amount of artificial
dissipation, it can also be worthwhile to develop schemes without numerical viscosity and to
understand the influence of time integration schemes on the stability of numerical methods,
both for (non-normal) linear and nonlinear systems. Additionally, the time integration schemes
have to be taken into account when further robustness properties of numerical methods are
considered, e.g. positivity preservation. The Minisymposium will consider several methods
• Deferred Correction Methods (DeC),
• Strong Stability Preserving Runge-Kutta Schemes (SSPRK),
• Relaxation Runge-Kutta Methods,
• Summation-by-parts in Time approaches (SAT-SBP in time),
including modified Patankar approaches introduced by Burchard et al. .
The aim of this Minisymposium is to bring together researchers with different backgrounds
working on these related topics and to facilitate an interchange of ideas and new developments
in this field.
4:00pm - 4:30pm
Relaxation deferred correction method and its application in Finite Element Discretization
1University Zurich, Switzerland; 2Johannes Gutenberg-University Mainz, Germany; 3INRIA Bordeaux, France
Recently, relaxation Runge-Kutta and multisteps schemes have been proposed by Ketcheson et al. which preserve the correct evolution of a function inside the numerical solution. In the context of hyperbolic conservation laws and balance laws, these relaxation time discretizations have been heavily applied together with semidiscrete entropy conservative/dissipative schemes resulting in fully discrete entropy conservative/dissipative schemes. Here, the splitting between space and time discretization has been done using a simple method of lines approach. However, in some context of finite element discretization one likes to avoid such space-time splitting like for example in the residual distribution (RD) framework where the high-order of the method would be lost.
In this context, the deferred correction (DeC) method seems to be a good alternative. Abgrall proposed a simplified DeC method and was able to combine it with the residual distribution (RD) framework to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix.
In this talk, we extend the relaxation approach to DeC and demonstrate first analogous results to relaxation RK (RRK) methods using now the DeC framework. We point out the connection to RRK methods and verify all our theoretical finding numerically. Afterwards, we apply the relaxation DeC method in the RD setting and analyze the resulting scheme analytically and numerically.
4:30pm - 5:00pm
On modified Patankar Schemes and oscillations: towards new stability definitions
1Inria Bordeaux - Sud-Ouest, France; 2Johannes Gutenberg Universität, Mainz, Germany; 3Fachbereich Mathematik und Informatik der Universität Münster, Germany
Modified Patankar (MP) schemes are linearly implicit ODE solvers for production destruction problems that guarantee unconditionally the positivity of the solutions and the conservation of the total quantities.
At the moment, different version of such schemes are available, some classes of Runge--Kutta (RK), strong stability preserving RK (SSPRK) and arbitrarily high order Deferred Correction (DeC) schemes.
The classical A-stability, L-stability properties cannot be directly computed on such schemes, for serveral reasons, inter alia, Dahlquist's equation is not a production destruction system and the mass matrix of the MP schemes depends on the variables. Moreover, we observed that for large timesteps these schemes may produce oscillations around the steady state solution.
We try to find new measures of stability for these type of schemes measuring different quantities of which the analytical behavior is known, e.g. dissipative Liyapunov functionals. This information can be computed analytically or numerically for the various parameters involved (scheme parameters, initial condition, time step size) in the procedure for simple problems. These measures show us which schemes are favorable to solve these problems and under which conditions (e.g. time step restrictions).
5:00pm - 5:30pm
Properties of Runge-Kutta-Summation-By-Parts Methods
1Lund University, Sweden; 2Linköping University, Sweden; 3Technion - Israel Institute of Technology
This talk aims at providing an overview of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators and their generalizations, herein called RK-SBP methods. Established knowledge about these methods has hitherto been derived by energy analysis. However, adopting the point of view of RK theory gives new insights into RK-SBP methods.
The theory of implicit Runge-Kutta methods is full of 'SBP-like' algebraic conditions. Here, we review them in relation to linear and nonlinear stability, stiffness and convergence. The importance of diagonal mass matrices and grid collocation for nonlinear problems are highlighted. Finally, we address precisely which SBP operators that can be associated with RK methods, namely those that are null-space consistent. The importance of positive definiteness of SBP operators is also discussed.
5:30pm - 6:00pm
Towards normal mode stability analysis for domain-based IMEX splitting of advection-diffusion problems
University of Kassel, Germany
Considering advection-diffusion problems, stiffness of the space-discretized system usually originates from the diffusion part. A common IMEX time integration approach thus consists in discretizing the diffusion terms implicitly while solving the advection terms in an explicit fashion. For certain combinations of space and time discretization, grid-independent stability may be achieved for this type of IMEX splitting. The reason for this favorable property is that the explicit discretization of advection is stabilized by the implicit treatment of diffusion. However, time steps become smaller for advection-dominated cases. In addition, for systems, grid-independent stability only holds if diffusion terms are present for each component. Unfortunately, this basically rules out grid-independent stability of advection-diffusion IMEX splitting for compressible viscous fluid flow.
On the other hand, domain-based IMEX splitting adopts implicit time discretization only in those parts of the fluid domain which are subject to the strongest time step restrictions. This approach is particularly useful in case of locally refined grids where the region of smallest cells is discretized implicitly. Stability and time step restrictions of domain-based IMEX splitting are difficult to determine due to the interface coupling between implicit and explicit region. For IMEX-DG schemes based on a combination of the forward and backward Euler scheme, strong L2 stability can be guaranteed under a CFL-like time step restriction, but higher order extensions of this result seem to be to difficult to obtain. In addition, this type of stability analysis fails to deliver reasonable step size conditions for the advection-dominated case.
Related problems are faced in the analysis of partioned schemes for coupled problems where the stability analysis of the time integration process is a challenging task as well. A particularly promising approach has been devised for conjugate heat transfer in fluid-structure interaction. Here, the technique of normal mode analysis has become an important tool to investigate the stability of interface treatments used to combine specific partitioned solvers. Originally, normal mode analysis has been used to determine stability conditions of finite difference schemes for non-periodic problems, thus extending classical von Neumann stability analysis.
For domain-based IMEX time integration of semi-discrete conservation laws, the splitting of interface terms additionally has to maintain certain conservation properties. Nonetheless, different splitting variants are basically possible and may have different performance with respect to stability. In this talk, we introduce the concept of normal mode stability analysis to domain-based IMEX splitting for advection-diffusion equations to investigate different interface treatments. Furthermore, numerical results using domain-based IMEX splitting based on partitioned RK time integration and DG space discretization are shown in particular for an advection-diffusion-reaction model of tumour angiogenesis and for compressible viscous fluid flow.
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