Conference Agenda

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Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 08:34:57pm CET

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Session Overview
Z7: time integrators, time-dependent problems
Monday, 12/July/2021:
4:00pm - 6:00pm

Session Chair: Herbert Egger
Virtual location: Zoom 7

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4:00pm - 4:20pm

On the Time Growth of the Error of the Discontinuous Galerkin Method for Advection-Reaction Problems

Václav Kučera1, Chi-Wang Shu2

1Charles University, Czech Republic; 2Brown University, USA

Gronwall's lemma is a basic tool in the estimation of various quantities for evolutionary problems. Its disadvantage is that its application leads to estimates that grow exponentially in time and cannot be used to produce e.g. uniform estimates on infinite time intervals. To overcome this limitation, finer methods of analysis tailored to the problem must be applied. In this talk we present error estimates for the discontinuous Galerkin (DG) method applied to linear nonstationary advection-reaction problems with a general advection field without any assumptions on the sign of its divergence, etc. We use a space-time exponential scaling argument to avoid the use of Gronwall's lemma and give an explicit construction of the necessary exponential scaling function. The result is an $L^\infty(L^2)$ error estimate that grows exponentially not in time, but with respect to the time particles carried by the flow spend in the spatial domain. If this particle life-time is uniformly bounded, one obtains uniform space-time error estimates even on infinite time intervals. The interpretation is that the presented technique allows to analyze the fundamentally Eulerian DG method in a Lagrangian manner, i.e. along characteristics of the flow field.

4:20pm - 4:40pm

Computing semigroups with error control

Matthew John Colbrook

University of Cambridge, United Kingdom

We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon>0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules and the adaptive computation of resolvents in infinite dimensions through rectangular truncations. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain $L^2(\mathbb{R}^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like $\exp(-cN/\log(N))$ for $N$ quadrature points, that remains stable as $N\rightarrow\infty$, and which is also suitable for infinite-dimensional operators. We discuss our results in the context of spectral methods on bounded and unbounded domains. Numerical examples are given, including: Schr\"odinger and wave equations on the aperiodic Ammann-Beenker tiling, complex perturbed fractional diffusion equations on $L^2(\mathbb{R})$, and damped Euler--Bernoulli beam equations.

4:40pm - 5:00pm

On the implicit time integration of an entropy conserving/stable discontinuous Galerkin method

Alessandra Nigro1, Andrea Crivellini1, Alessandro Colombo2

1Marche Polytechnic University, Italy; 2University of Bergamo, Italy

A faithful representation of entropy evolution is a very attractive property for high-fidelity iLES/uDNS simulations of turbulent flows. To this end, in recent years, several numerical fluxes have been proposed in literature [1,2,3]. Additionally, since these fluxes produce very strong oscillations in the presence of shocks, or if the mesh is too much coarse [2], novel entropy stable fluxes have been proposed in order to overcome this drawback, adding to the entropy conservative ones, that are typically centered fluxes, a dissipation that enforces the production of entropy for discontinuous solutions [1,2]. These semi-discrete entropy conserving/stable numerical schemes have been investigated in several papers, see for example the review article by Chen and Shu [4]. However, the question of entropy behaviour of explicit and implicit time integration schemes is still an open question. In fact, even if there is an evident non-negligible effect of time integration on the entropy properties, there are few papers that include theoretical aspects or an extensive analysis of numerical experiments about this topic [5,6].

The aim of this paper is to assess the effect of the time-integration on the entropy conservation, and to contribute to the development of a high-order fully-discrete entropy conserving/stable scheme for the solution of unsteady inviscid flows. To this end, we consider a discontinuous Galerkin (dG) method based on the set of entropy variables, over-integration of the discretized terms (using very high-order accurate Gauss quadrature rules), and entropy conserving/stable numerical fluxes [1,2,3]. Differently from the space/time scheme presented in [7], here the more commonly used method of lines is considered, and therefore several high-order accurate linearly-implicit [8] and implicit [5,9] time integration schemes are examined (Rosenbrock, BDF, MEBDF, CN). Among them, we will show in particular different CN-type time discretizations, inspired by the generalized Cranck-Nicolson scheme presented in [5], that has been adapted to our numerical framework, i.e. the high-order dG method with working variables different from the conservative ones considered in [5].

The accuracy and the properties of the fully-discrete scheme are assessed in a series of numerical experiments performed with well known test-cases that have an exact analytical solution. We will show how high-order time integration schemes effectively reduce the entropy production/dissipation since they quickly converge to an entropy conserving solution (if the discretization is entropy conserving in space). This means that, from the practical point of view, the entropy production of an entropy stable space discretization can easily dominate the temporal errors, even for relatively large time step sizes. Furthermore, we will show that one of the CN-type time discretization here proposed is almost entropy conserving, since a very small entropy error appears only when a very coarse space discretization is used. Finally, same relevant issues related to the use of entropy conserving fluxes are addressed, like the kinetic energy conservation of the fully-discrete scheme when long-time simulations are performed, and the comparison between the accuracy provided by the entropy and the primitive set of variables.


[1] F. Ismail, P. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks, Journal of Computational Physics 228 (15) (2009) 5410–5436.

[2] P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Communications in Computational Physics 14 (5) (2013) 1252–1286.

[3] H. Ranocha, Entropy Conserving and Kinetic Energy Preserving numerical methods for the Euler equations using Summation-by-Parts Operators, in: S. J. Sherwin, D. Moxey, J. Peiró, P. E. Vincent, C. Schwab (Eds.), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Springer International Publishing, Cham, 2020, pp. 525–535.

[4] T. Chen, C.-W. Shu, Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes, CSIAM Transactions on Applied Mathematics 1 (1) (2020) 1–52

[5] P.G. Lefloch, J.M. Mercier, C. Rohde, Fully-discrete,entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal. 40, 1968–1992 (2002).

[6] A. Gouasmi, S.M. Murman, K. Duraisamy, Entropy Conservative Schemes and the Receding Flow Problem, Journal of Scientific Computing 78 (2019) 971–994.

[7] J. Barth, Numerical Methods for Gasdynamic Systems on Unstructured Meshes, Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, pp. 195–285

[8] F. Bassi, L. Botti, A. Colombo, A. Ghidoni, F. Massa, Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows, Computers and Fluids 118 (2015) 305–320.

[9] A. Nigro, A. Ghidoni, S. Rebay, F. Bassi, Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows, International Journal for Numerical Methods in Fluids 76(9) (2014) 549-574.

5:00pm - 5:20pm

Error Inhibiting Methods with Post-Processing for Ordinary Differential Equations.

Adi Ditkowski1, Sigal Gottlieb2, Zachary J. Grant3

1Tel Aviv Univerrsity, Israel; 2University of Massachusetts Dartmouth,; 3Oak Ridge National Laboratory, Oak Ridge TN, USA

Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multistep methods, and more broadly general linear methods all have a global error of the same order as the local truncation error. In prior work, we investigated the interplay between the local truncation error and the dynamics of the schemes. This study enabled us to construct error inhibiting methods that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error.

In this talk, we extend our error inhibiting framework to include a broader class of time-discretization methods that allows an exact calculation of the leading-error term. This knowledge enables us to post-process this term to obtain a solution that is two orders higher than expected from truncation error analysis. We define sufficient conditions that result in a desired form of the error term and describe the construction of the post-processor.

Several new, high-order, explicit and implicit methods, and SSP and IMEX methods that have this property are given and tested on various ordinary and partial differential equations. We demonstrate that these methods indeed provide a solution that is two orders higher than expected from truncation error analysis alone.

5:20pm - 5:40pm

A Relaxed Multirate Integrator for Hyperbolic equations

Shinhoo Kang, Emil Constantinescu

Argonne National Laboratory, United States of America

A grid refinement is a common technique used to capture fine structures. For numerical stability, explicit time integrators suffer from the timestep size restriction due to geometrically induced stiffness. Multirate methods lower the computational burden by using small timesteps only for refined regions and large timesteps for coarser regions. Meanwhile, nonlinear interaction can turn a smooth solution into a sharp gradient solution, which also can trigger numerical instability. To amend the issues, we develop a relaxed multirate time-stepping scheme for hyperbolic conservation laws that not only alleviate computationally taxing explicit time integrators for stiff problems but also provide favorable properties that include total mass conservation and entropy conservation (or entropy dissipation). Numerical experiments are conducted to show the stability and accuracy of the proposed methods for one-dimensional hyperbolic equations in the context of entropy conservative (or entropy stable) spatial discretization.

5:40pm - 6:00pm

Exponential Time Integrators for Numerical Weather Prediction

Greg Rainwater1, Kevin Viner2, Alex Reinecke2

1American Society of Engineering Education, Monterey, California, United States of America; 2Naval Research Laboratory, Monterey, California, United States of America

Recent advancements in evaluating matrix-exponential functions have opened the doors for the practical use of exponential time-integration methods in numerical weather prediction (NWP). The success of exponential methods in shallow-water simulations has led to the question of whether they can also be beneficial in a 3D atmospheric model. In this talk we use exponential time integration methods in a fully compressible deep-atmosphere non-hydrostatic global spectral element model (NEPTUNE-The Navy Environmental Prediction System Utilizing a Nonhydrostatic Engine) and demonstrate their superior stability and high accuracy on set of idealized test cases. Performance comparisons with other time integrators reveals that exponential methods do have a place in NWP. Potential avenues for further improving the efficiency of exponential methods will also be discussed.

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