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Fredag 17 maj 2024, kl. 13:15-14:15, Jan Glaubitz, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology

Titel: Beyond the conventional: Summation-by-parts operators for general function spaces

Sammanfattning: Numerically solving time-dependent partial differential equations (PDEs) is essential across many scientific and engineering disciplines. Preserving critical structures of PDEs when discretizing them improves the accuracy and robustness of numerical methods, which is particularly important for advection-dominated problems. Summation-by-parts (SBP) operators have emerged as a widely utilized tool in this endeavor, mirroring integration-by-parts at the discrete level. This allows us to transfer results, such as stability estimates, from the continuous to the discrete level.

Traditional SBP operators start from the assumption that the PDE’s solution can be well approximated by polynomials. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. The benefits of nonpolynomial approximation spaces are well established, including solvers based on kernels, rational functions, and artificial neural network approximations.

In this talk, I aim to develop a theory of function-space SBP (FSBP) operators, which are SBP operators designed for general function spaces beyond just polynomials. These FSBP operators maintain the desired structure-preserving properties of existing polynomial SBP operators while allowing for greater flexibility by applying to a broader range of function spaces.

This talk is based on joint work with Anne Gelb (Dartmouth College, USA), Jan Nordström (Linköping University, Sweden & University of Johannesburg, South Africa), and Philipp Öffner (TU Clausthal, Germany).

Keywords: Advection-dominated PDEs, stability, summation-by-parts operators, non-polynomial function spaces.

Onsdag 8 februari professor Charis Harley, University of Johannesburg, Sydafrika

Plats: Kompakta Rummet, ingång 23, B-huset. Tid: 13.15-14.15

Titel: A provably stable and conservative hybrid scheme for the singular Frank-Kamenetskii equation

Abstract: We consider the Frank-Kamenetskii partial di erential equation as a model for combustion in Cartesian, cylindrical and spherical geometries. Due to the presence of a singularity in the equation stemming from the Laplacian operator, we consider a specific conservative continuous formulation thereof, which allows for a discrete energy estimate. Furthermore, we consider multiple methodologies across multiple domains. On the left domain, close to the singularity, we employ the Galerkin method which allows us to integrate over time appropriately, and on the right domain we implement the finite difference method. We also derive a boundary condition that removes a potentially artificial boundary layer. The summation-by-parts (SBP) methodology assists us in coupling these two numerical schemes at the interface, so that we end up with a provably stable and conservative hybrid numerical scheme. We provide numerical support for the theoretical derivations and apply the procedure to a realistic case.

Torsdag 30 januari 2020, Mark H. Carpenter, NASA Langley Research Center, USA

Titel: Nonlinear Stability and the Discrete Equations of Fluid Mechanics

Sammanfattning: A high-level overview of the Summation-By-Parts (SBP) entropy stability literature is presented. Nonlinearly (entropy) stable discretizations of arbitrary order exist for the compressible Navier-Stokes (NS) equations for diagonal norm, tensor-product and multi-dimensional, summation-by-parts (SBP) operators. Recent developments are discussed:

1) Curvilinear brick elements,
2) generalized SBP extensions (triangles, prisms, tets),
3) nonconforming interfaces that enable hp-refinement,
4) arbitrary Lagrangian-Eulerian operators,
5) staggered operators resembling classical FEM,
6) extension to fully discrete operators (SBP in time),
7) nonlinearly stable boundary conditions,
8) dissipation: finite-difference and spectral collocation WENO operators,
9) other related equations.

Impediments are identified that currently hinder/preclude a general and complete nonlinear stability theory for arbitrary discretizations and equations.

Onsdag 11 september 2019, Thomas Hagstrom, Southern Methodist University, USA

Titel: Radiation boundary conditions for waves: extensions and open problems

Sammanfattning: The radiation of energy to the far field is a central feature of wave physics. As such efficient, convergent domain truncation algorithms are a necessary component of any wave simulation software. For the scalar wave equation and equivalent systems such as Maxwell’s equations in a uniform dielectric medium, Complete Radiation Boundary Conditions (CRBC), which are optimized local radiation boundary condition sequences, provide a satisfactory solution: spectral convergence, rapid parameter selection based on sharp a priori error estimates, and effective corner/edge closures. Issues that arise in extending the method to more general problems include the treatment of so-called reverse modes, waves whose group and phase velocities are misaligned relative to the normal direction at the radiation boundary, as well as problems with inhomogeneities and nonlinearities. Here we will explore a number of ideas for constructing reliable and efficient domain truncation algorithms in these more difficult settings:

  • Modifications of the CRBC recursions to handle reverse modes - this approach is successful for problems with single wave families;
  • Rigorous a priori error analysis and optimization of ad hoc damping/stretching layers;
  • Application of kernel compression/reduced order modeling to nonlocal formulations.

Specific applications to dispersive models of electromagnetic waves, the elastic wave equation, as well as waves in inhomogeneous media will be given.

Tisdag 3 september 2019, Byron Jacobs, University of the Witwatersrand, Sydafrika

Titel: An overview of Numerical Solution Techniques for Fractional Order Problems

Sammanfattning: In this presentation I aim to give an overview of some numerical techniques for fractional differential equations as well as some justifications for fractional derivatives as modelling tools. This work is primarily limited to `parabolic' type equations: sub-diffusion, time-fractional Fokker-Planck, and some Fractional ODE work. A high-level analysis of the benefits and shortcomings for each method is discussed. References are provided for examples, rigorous numerical analysis and in-depth explanation/derivation of each method.

Onsdag 28 februari 2018, Jens Berg, Uppsala universitet

Titel: Neural networks and partial differential equations

Sammanfattning: Artificial neural networks are behind much of the recent progress in machine learning and artificial intelligence. There are a large variety of neural networks depending on the area of application. Most commonly known are the convolutional neural networks for image processing, and recurrent neural network for natural language processing. It is easily forgotten that the most basic neural network, the feedforward neural network, is an efficient global function approximator. In this talk, we will use basic neural networks as basis functions for approximations of the solution to partial differential equations, inverse coefficient estimation, and data driven discovery of linear and non-linear PDE models.

Tisdag 19 december 2017, Mark H. Carpenter NASA Langley Research Center, Virginia, USA

Titel: Entropy stability for the compressible Navier-Stokes equations: operator generalizations and the non-conforming interfaces

Sammanfattning: Nonlinearly stable (entropy stable) discretizations of arbitrary order can be constructed for the compressible Navier-Stokes (NS) equations for all diagonal norm, tensor-product, summation-by-parts (SBP) opera-tors. The NS equations are discretized in strong conservation form, and a novel choice of nonlinear fluxes ensures pointwise conservation of mass, momentum, energy and a nonlinear entropy function that guarantees L2 stability of the (semi-)discrete solution. The stability estimates are sharp and do not rely on common assumptions of “integral exactness”, or “added dissipation”. The discrete operators are fully consistent with the Lax-Wendroff theorem. Thus, captured shocks converge to weak solutions provided physical dissipation is sufficient at shocks.

Initial developments (circa 2013) focused on the nonlinear stability of 4th and 6th-order finite-difference operators, and their extension to WENO operators on structured curvilinear meshes. Shortly thereafter (2014-15), entropy stability analysis was used to develop spectral collocation operators on curvilinear brick elements. Subsequently, nonlinear stability theory for SBP operators has rapidly evolved. Recent progress has been achieved on 1) multiple collocation strategies for hexahedral elements, 2) nonlinear stable boundary conditions, 3) WENO stencil biasing for spectral collocation operators, 4) extension to triangular and tetrahedral elements, 5) nonconforming hexahedral element interface operators that allow for h- and p-refinement, to name a few.

Herein, a high-level overview of the SBP entropy stability literature is given. Then, recent progress is reported on developing entropy-stable (SS) discontinuous spectral collocation formulations for hexahedral elements. This effort extends previous work on entropy stability to include p-refinement at curvilinear non-conforming interfaces. A generalization of existing entropy stability theory is required to accommodate the nuances of the nonconforming curvilinear coupling. The entropy stability of the compressible Euler equations is demonstrated using the newly developed operators. Canonical test cases are used to demonstrate the efficacy of the new operators.

Tisdag 6 december 2016, Per Pettersson, Uni Research, University of Bergen, Norge

Titel: Uncertain transport problems in large-scale carbon storage: stochastic basis reduction and robust numerical discretization

Sammanfattning: Large-scale storage of CO2 in subsurface reservoirs has been identified as an essential means to reduce emission of greenhouse gases to the atmosphere. The North Sea contains potential subsurface storage sites where significant amounts of CO2 can be injected in the next 50 years, but the fate of the injected CO2 over the following hundreds of years is uncertain due to the migration of the gas in the underground. Numerical simulation of CO2 in subsurface storage reservoirs is therefore of great interest, but it is computationally demanding due to the large ranges of spatial and temporal scales. In addition, the material input parameters are unknown due to lack of data. Efficient representation of these uncertainties is challenging due to the hyperbolic nature of the governing equations. Even a relatively simple deterministic problem formulation is challenging to solve numerically due to non-convex and discontinuous flux functions.

In this talk we present a reduced-order stochastic Galerkin method by means of locally discarding insignificant chaos modes. The method's performance is demonstrated on the stochastic Buckley-Leverett equation and we discuss suitable discretization techniques.

Tisdag 15 november 2016, Magnus Svärd, University of Bergen, Norge

Titel: Entropy stability, convergence and the Navier-Stokes equations

Sammanfattning: In this talk, I will present the theory for entropy stability and other numerical a priori estimates. Furthermore, I will discuss the difficulties encountered when proving convergence for the non-linear Euler and Navier-Stokes equations.

Tisdag 26 april 2016, David A. Kopriva, Florida State University, USA

Titel: Spectral Methods That Work

Sammanfattning: Spectral methods are high order orthogonal polynomial expansion approximation techniques for partial differential equations that promise rapid, even exponential, convergence. Approximations for hyperbolic equations exhibit low dissipation and dispersion errors, making them especially suitable for wave propagation problems. Spectral element versions enable the methods to be used in complex geometries. Unfortunately, the high order and low dissipation also make the methods sensitive to aliasing instabilities, even for linear equations. In this talk, I show how to develop provably stable discontinuous Galerkin spectral element methods with an eye towards mimicking the energy properties of the original PDE. In other words, spectral methods that work.


Arkiv 2011-2015

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