08:30
MS22: Structure-preserving discretization methods II: Variational methods for physics compatible discretizations (Part 2)
Chair: Marc Gerritsma
08:30
25 mins
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Semi-Lagrangian Reconstruction Finite Elements for Advection-dominated Problems with Rough Data
Konrad Simon
Abstract: Long time scales in climate simulations, e.g., in simulations of paleo climate, require coarse grids due to
computational constraints. Coarse grids, however, leave important smaller scales unresolved. Thus small scale processes that significantly influence the resolved scales need to be taken care of by different means. Such subgrid processes include (slowly) moving land-sea interfaces or ice shields as well as flow over urban areas or biogeochemistry. State-of-the-art dynamical cores represent the influence of subscale processes typically via subscale parametrizations and often employ a rather heuristic coupling of scales.
We aim to improve the mathematical consistency of the upscaling process that transfers information from the subgrid to the coarse prognostic and diagnostic quantities (and vice-versa). We investigate new bottom-up techniques for advection-dominated problems whose main motivation are climate simulations~[Lauritzen et al., 2011]. Our tools are based on ideas for multiscale finite element methods for elliptic problems that play a role, in oil reservoir modeling and porous media in general~[Efendiev et al., 2009; Graham et al., 2012]. These ideas, however, fail in advection-dominated scenarios (which are typical for flows encountered in climate models) since they are not based on a suitable decomposition of the computational domain.
We present new Garlerkin based ideas to account for the typical difficulties in climate simulations. Our idea is based on a previous work that employs a change of coordinates based on a coarse grid characteristic transform induced by the advection term to make its effect on coarse scales milder. This also accounts for appropriate subgrid boundary conditions for the multiscale basis functions. Boundary conditions are essential for such approaches. This is the starting point of a set of semi-Lagrangian techniques that locally in time reconstruct subgrid variability in a Galerkin basis through many local inverse problems. We discuss extensions and drawbacks of this approach and present examples with rapidly varying coefficients on several scales.
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08:55
25 mins
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Symplectic discretization of an augmented Lagrangian method for optimal control
Jason Frank, Xin Liu, Sarah Gaaf
Abstract: Direct numerical discretization---using a Runge-Kutta method---of the Lagrange multiplier form of the constrained optimization form of optimal control problems gives rise to a symplectic partitioned Runge-Kutta pair for the Pontryagin maximum principle (Sanz-Serna, SIAM Review 2016). In this talk we demonstrate how symplectic discretization of an augmented Lagrangian method leads to a globally convergent forward-backward iteration for optimal control problems.
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09:20
25 mins
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Arbitrary order variational integrator for Hamiltonian systems
Artur Palha Palha, Marc Gerritsma
Abstract: Traditionally, the methods developed for the discrete solution of ordinary differential equations derive from finite difference principles and Taylor series expansions. This approach focuses on minimizing the local truncation error associated with the discretization. No matter how small this truncation error is, it is always finite, which might lead to a continuously growing global error. Over a sufficiently long time interval, the global error will inevitably reach unacceptable values and eventually the numerical solution will no longer resemble the analytical one. Due to this intrinsic local truncation error of the discretization process, one wishes to develop methods that are able to preserve the same qualitative properties as the continuous system. This will enable acceptable long time integration.
For these reasons, there is a growing interest in the conservation of invariants when numerically solving a system of ordinary differential equations. This is particularly relevant for systems that contain invariants, like Hamiltonian systems. Methods that exactly preserve these quantities in time are known as \emph{geometric integrators}. In this work we use the mimetic framework \cite{Kreeft2011,Palha2014} together with a variational formulation to solve Hamiltonian systems of equations. It will be shown that this formulation is closely related to a Least-Squares finite element method in time. Additionally, we will also show that the quadrature rule used to compute the inner products will generate two integrators with different properties. Moreover, in terms of differential geometry, these two integrators will differ only on the discretization of the Hodge-$\star$ operator. It is shown that the one based on a canonical Hodge-$\star$ results in a symplectic integrator, whereas the one based on a Galerkin Hodge-$\star$ results in an energy preserving integrator. A set of numerical tests confirms these theoretical results.
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09:45
25 mins
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Geometric Discontinuous Galerkin Methods for Fluids and Plasmas
Michael Kraus
Abstract: Most conservative problems in fluid dynamics, plasma physics as well as many other branches of science and engineering have the form of hyperbolic conservation laws that inhibit a Lagrangian and/or Hamiltonian structure. That is their dynamical equations can be obtained from an action principle or a Poisson bracket and a Hamiltonian functional, typically the total energy of the system.
Non-conservative problems are usually composed of a conservative (Lagrangian or Hamiltonian) and a dissipative part.
In both cases, it is important to preserve the structure of the conservative part in the course of discretisation in order to obtain stable numerical schemes that deliver accurate and reliable simulation results.
We discuss Lagrangian and Hamiltonian structure-preserving discretisation approaches based on high-order Discontinuous Galerkin Spectral Element Methods (DGSEM). We show how these approaches relate to and generalise known energy-stable schemes based on split-forms and summation-by-parts properties.
The inviscid Burgers equation and the compressible Euler equations serve as main examples. Generalisations to other important fluid and plasma systems are sketched.
A remarkable property of the proposed approach is that exact mass, momentum and energy conservation can be achieved even if the system of equations is not cast in conservative form and momentum and energy do not explicitly appear as variables.
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