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: 4th Dec 2022, 07:31:59pm CET
MS 1b: 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.
11:50am - 12:20pm
Design of an adaptive time-stepping method by balancing discretization errors in an implicit compressible flow solver
1State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, PRA; 2Department of Aeronautics, Imperial College London, UK
In an implicit compressible flow solver based on Diagonally implicit Runge-Kutta method and Jacobian-free Newton Krylov method, there are parameters such as temporal accuracy, time step, Newton tolerance, tolerance of iterative linear solver and preconditioner related parameters. Proper choosing these parameters, which have significant influences on the accuracy, efficiency and stability, is essential for developing a robust general purpose solver. A balanced adaptive time-stepping strategy is implemented in the implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations. The temporal and spatial errors are estimated using an embedded Runge-Kutta scheme and a higher-order spatial discretization. Based on the idea of constructing proper relations between the temporal, spatial and iterative errors, an adaptive strategy is designed for determining the time step and the Newton tolerance. These parameters maintain temporal accuracy of the solver and are relatively efficient by avoiding excessively small time step and Newton tolerance choices. The design is tested in different types of cases, including isentropic vortex convection, steady-state flow past a flat plate, Taylor-Green vortex and turbulent flow over a circular cylinder, to illustrate its performance and its generality. The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efficiency is obtained for unsteady and steady, well-resolved and under-resolved simulations.
12:20pm - 12:50pm
An efficient all-speed scheme for the Euler equations
University of Mainz, Germany
Stable and efficient time integration schemes for the Euler equations are especially important when the material velocity is much slower than the acoustic waves. In these regimes, which are characterized by small Mach numbers, a full resolution of all waves requires very small time steps, while usually one is mainly interested in the slow dynamics of the material wave which would allow for a much larger time step. The use of explicit schemes requires a Mach number dependent time step that vanishes if the Mach number tends to zero. Therefore using explicit schemes in such regimes is very costly. In this talk, we focus on the resolution of the slow material wave using an implicit-explicit (IMEX) framework with a Mach number independent time stepping.
As it is typical in the IMEX setting, fast scales are integrated implicitly which in case of the Euler equations usually results in having to solve a non-linear implicit system. To avoid this, we use a Suliciu-type relaxation approach that leads to an implicit part consisting only of a scalar linear equation and simplifies at the same time the construction of an approximate Riemann solver for the explicit part. The scheme is stable and computationally efficient for all Mach regimes and recovers the correct asymptotic limit which yields the incompressible Euler equations for suitable initial data.
In this talk, we describe in detail the second order finite volume scheme, explain the relation to known IMEX approaches and give an outlook on the construction of higher order extensions and the treatment of gravitational source terms.
12:50pm - 1:20pm
Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows
1Michigan State University, United States of America; 2Brown University, United States of America; 3University of Science and Technology Beijing, China
In this talk, we will present a family of second-order and third-order modified Patankar Runge-Kutta (MPRK) methods, which are conservative and unconditionally positivity-preserving, for production-destruction equations. We derive necessary and sufficient conditions to obtain the designed order of accuracy. This ordinary differential equation solver is then extended to solve a class of semi-discrete schemes for PDEs. Combining this time integration method with the positivity-preserving finite difference weighted essentially non-oscillatory (WENO) schemes, we successfully obtain a positivity-preserving WENO scheme for non-equilibrium flows. Various numerical tests are reported to demonstrate the effectiveness of the methods.
1:20pm - 1:50pm
Third-Order Paired Explicit Runge-Kutta Schemes for High-Order Spatial Discretizations
Concordia University, Canada
In this paper, we expand the family of Paired Explicit Runge-Kutta methods, referred to as P-ERK schemes, to third-order accuracy in time. The P-ERK approach is particularly appealing for the solution of locally stiff systems of Partial Differential Equations (PDEs) to reduce computational cost. Previously, it was shown that P-ERK family members with different numbers of derivative evaluations can be applied to different regions of the domain, based on local stiffness requirements, so the global time-step to be taken as large as possible. However, the original family of P-ERK schemes is limited to second-order accuracy. In this study, we introduce a new family of P-ERK methods with third-order accuracy and optimize them specifically for high-order flux reconstruction spatial discretizations. We verify that these methods achieve their designed order of accuracy for an isentropic vortex case with arbitrary combinations of schemes. We then demonstrate that simulations with the third-order P-ERK family can achieve significant speedup factors compared with classical third-order Runge-Kutta methods. These speedup factors are obtained for laminar and turbulent flow over an SD7003 airfoil and turbulent tandem spheres simulations. These results demonstrate the P-ERK schemes are a suitable approach for accelerating high-order accurate simulations of unsteady turbulent flows.
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