BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Kurgano (Southern University of Science and Technology) DTSTART:20210823T130000Z DTEND:20210823T135000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/1 DESCRIPTION:Title: Well-Balanced Schemes via Flux Globalization\nby Alexander Kurgano (Southern University of Science and Technology) as part of CMO-Bound-Pres erving Space and Time Discretizations\n\n\nAbstract\nWe consider hyperboli c systems of balance laws. Our goal is to develop well-balanced numerical methods\, which respect a delicate balance between the flux and source ter ms and are thus capable of exactly preserving (some of the) physically rel evant steady-state solutions of the studied systems.\nI will introduce a g eneral approach of constructing well-balanced schemes via a flux globaliza tion ap- proach: The source terms are incorporated into the fluxes. This r esults in the hyperbolic system of conser- vation laws with global fluxes. Such systems can be then integrated using Riemann-problem-solver-free num erical methods. I will show several recent non-straightforward application s of these well-balanced schemes.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/1/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ketcheson (King Abdullah University of Science & Technology) DTSTART:20210823T135000Z DTEND:20210823T144000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/2 DESCRIPTION:Title: Time discretizations that accurately evolve a conserved or dissipated functional\nby David Ketcheson (King Abdullah University of Science & Technology) as part of CMO-Bound-Preserving Space and Time Discretizations \n\n\nAbstract\nMany mathematical models are equipped with an energy that is conserved or an entropy that is known to change monotonically in time. Integrators that preserve these properties discretely are usually expensiv e\, with the best-known examples being fully-implicit Runge-Kutta methods. I will present a modification that can be applied to any integrator in or der to preserve such a structural property. The resulting method can be fu lly explicit\, or (depending on the functional) may require the solution o f a scalar algebraic equation at each step. I will present examples to sho w the effectiveness of these “relaxation” methods\, and their advantag es over fully implicit methods or orthogonal projection. Examples will inc lude applications to compressible fluid dynamics\, dispersive nonlinear wa ves\, and Hamiltonian systems.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Gerardo Hernández Dueñas (Universidad Nacional Autonoma de Mexic o) DTSTART:20210823T151000Z DTEND:20210823T160000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/3 DESCRIPTION:Title: Two-layer Shallow-water Flows along Channels with Arbitrary Geometry\nby Gerardo Hernández Dueñas (Universidad Nacional Autonoma de Mexico ) as part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbst ract\nWe present a new high-resolution\, non-oscillatory semi-discrete cen tral scheme for one-dimensional two- layer shallow-water flows along chann els with arbitrary cross sections and bottom topography. The scheme extend s existing central semi-discrete schemes for hyperbolic conservation laws and it enjoys two properties crucial for the accurate simulation of shallo w-water flows: it preserves the positivity of the water height\, and it is well balanced\, i.e.\, the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gr adients. The system is integrated in time using a second order Strong Stab ility Preserving Runge-Kutta scheme. Along with a detailed description of the scheme and proofs of these two properties\, we present several numeric al experiments that demonstrate the robustness of the numerical algorithm. This is joint work with Jorge Balbas.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Manuel Quezada de Luna (Universidad Nacional Autonoma de MexicoKin g Abdullah University of Science and Technology) DTSTART:20210823T160000Z DTEND:20210823T165000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/4 DESCRIPTION:Title: Bound-preserving flux limiting for high-order Runge-Kutta time discret izations of convection dominated problems\nby Manuel Quezada de Luna ( Universidad Nacional Autonoma de MexicoKing Abdullah University of Science and Technology) as part of CMO-Bound-Preserving Space and Time Discretiza tions\n\n\nAbstract\nWe first propose new flux limiters that have the stru cture of a flux-corrected transport (FCT) algo- rithm. These limiters do n ot depend on the time step size and\, therefore\, are applicable to spatia l semi- discretizations. Our limiters contain a user defined parameter tha t improves accuracy at a cost of a more restrictive CFL-like condition\, f or explicit implementations. To test the accuracy properties of these lim- iters\, we consider a spatial semi-discretization of Burgers’ equation based on arbitrarily high-order WENO reconstructions and use the limiters to impose global bounds.\nUsing the same WENO reconstructions\, we obtain a full discretization based on Runge-Kutta methods. This scheme is arbitra rily high-order and ‘essentially’ non-oscillatory but does not preserv e the maximum principle. To guarantee the maximum principle we combine the fluxes of the high-order scheme with those of a low-order method based on forward Euler and Local Lax-Friedrichs (LLF) fluxes. We obtain anti-diffu sive fluxes that combine corrections in space-and-time to the low-order sc heme. We use our proposed limiters to guarantee the solution is maximum pr inciple preserving. Finally\, we present a similar methodology using an ar bitrarily high-order Singly Diagonal RK method combined with a low-order s cheme based on backward Euler and LLF fluxes. The implicit scheme is maxim um principle preserving for time steps of any size.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Stechmann (King Abdullah University of Science and Technolo gy) DTSTART:20210823T182000Z DTEND:20210823T191000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/5 DESCRIPTION:Title: Numerical challenges in atmospheric dynamics with moisture and clouds< /a>\nby Samuel Stechmann (King Abdullah University of Science and Technolo gy) as part of CMO-Bound-Preserving Space and Time Discretizations\n\nAbst ract: TBA\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Martin Berzins (University of Utah) DTSTART:20210823T191000Z DTEND:20210823T200000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/6 DESCRIPTION:Title: Positive and/or Bounded Interpolation in time Integration - Applicatio ns and Theory\nby Martin Berzins (University of Utah) as part of CMO-B ound-Preserving Space and Time Discretizations\n\n\nAbstract\nWhile positi vity and boundedness preservation plays a key role in convection dominated problems it also plays a key in a number of situations in PDEs where disc rete interpolation is required. These situations include (i) Mapping from physics to dynamics grids in weather codes (ii) discrete remappings in ada ptive meshing (iii) particle to mesh remapping in particle in call an mate rial point methods (iv) solution recreation in AMR codes to address comput e node failures.\nTime integration error analysis shows how interpolation errors must be controlled to avoid polluting the main calculation and the connection with stage errors in Runge-Kutta methods in he case of particle methods.\nFor interpolation itself we provide a simple constructive proof for an adaptive algorithm tensor-product grids on arbitrary spacings to p reserve boundedness and positivity and show results for cases (i) and (iii ) above. Reference is also made to a longer more comprehensive proof.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Andreas Rupp (Lappeenranta-Lahti University of Technology LUT) DTSTART:20210824T130000Z DTEND:20210824T135000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/7 DESCRIPTION:Title: Variations of enriched Galerkin methods for the linear advection equat ion\nby Andreas Rupp (Lappeenranta-Lahti University of Technology LUT) as part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstr act\nWe interpret the enriched Galerkin (EG) method as generalization of s tandard finite elements (contin- uous Galerkin\, CG) and of the discontinu ous Galerkin (DG) method by combining the continuous and the discontinuous trial spaces of CG and DG\, and by using the DG bilinear and linear forms .\nThen\, we introduce algebraic flux correction schemes for the standard enriched (P1⊕P0 and Q1⊕P0) Galerkin discretizations of the linear adve ction equation. Here\, the piecewise-constant component stabilizes the con tinuous Galerkin approximation without introducing free parameters. Howeve r\, violations of discrete maximum principles are possible in the neighbor hood of discontinuities and steep fronts. To keep the cell averages and th e degrees of freedom of the continuous P1/Q1 component in the admissible r ange\, we limit the fluxes and element contributions\, the complete remova l of which would correspond to first-order upwinding.\nFinally\, we discus s a further generalization of the enriched Galerkin method. The key featur e of this step is an adaptive two-mesh approach that\, in addition to the standard enrichment of a conforming finite element discretization via disc ontinuous degrees of freedom\, allows to subdivide selected (e.g. troubled ) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. Here\, we prove stability and sharp a pr iori error estimates for a linear advection equation under appropriate ass umptions.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Remi Abgrall (Universität Zürich) DTSTART:20210824T135000Z DTEND:20210824T144000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/8 DESCRIPTION:Title: On the notion of conservation for hyperbolic problem\nby Remi Abgr all (Universität Zürich) as part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstract\nSince the celebrated Lax Wendroff convergen ce theorem\, published in 1960 in CPAM\, every one knows what should be th e structure of a finite volume/finite difference scheme so that one can ha ve a reasonable hope of convergence towards the ‘true’ entropy solutio n. The proof can easily be adapted to schemes like the discontinuous Galer kin ones\, thought it becomes a bit less clear. There are however many sch emes that does not fit clearly in that framework: continuous finite elemen t methods\, for example. Though it is relatively easy to prove a variant o f the Lax Wendroff for them\, this does not answer the question of the eng ineer: show me explicitely the flux. There are other questions related to conservation. We all know that it is forbiden by the Law to discretize the non conservative version of a conservative system (for example\, the Eule r equations in primitive variables)\, and there are many counter examples. However\, to which extend is that statement true? If one has an additiona l conservation law satisfied by the system (for example entropy conservati on for smooth solutions\, or kinetic momentum preservation)\, how can we m odify a given ‘good’ scheme so that the modified one will satisfy all constraints?\n\nIn this talk\, which sumarizes [1\,2\,3\,4\,5\,6]\, I will try to show the boundaries of these statements\, and provide example of s chemes\, some already known\, some more recent\, that contradict\, in some sense\, the standard beliefs. But not too much.\n\nReferences\n1. R. Abgr all. Some remarks about conservation for residual distribution schemes. Co mputational Meth-\nods in Applied Mathematics\, 18(3):327–350\, 2018.\n2 . R. Abgrall and S. Tokareva. Staggered grid residual distribution scheme for Lagrangian hydrodynamics.\nSIAM J. Scientific Computing\, 39(5):A2317 –A2344\, 2017.\n3. R. Abgrall. A general framework to construct schemes satisfying additional conservation relations\,\napplication to entropy con servative and entropy dissipative schemes. J. Comput. Phys\, 372(1)\, 2020 .\n4. Nathaniel Morgan Rémi Abgrall\, Konstantin Lipnikov and Svetlana To kareva. Multidimensional Stag-\ngered Grid Residual Distribution Scheme fo r Lagrangian Hydrodynamics. SIAM J. Sci. Comput.\,\n42(1):A343–A370\, 20 20.\n5. R. Abgrall. A combination of Residual Distribution and the Active Flux formulations or a new class\nof schemes that can combine several writ ings of the same hyperbolic problem: application to the 1D\nEuler equation s\, 2021. https://arxiv.org/abs/2011.12572\n6. R. Abgrall\, P. Öffner\, a nd H. Ranocha. Reinterpretation and Extension of Entropy Correction Terms\ nfor Resi\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Hennes Hajduk (TU Dortmund) DTSTART:20210824T151000Z DTEND:20210824T160000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/9 DESCRIPTION:Title: Property-preserving discontinuous Galerkin methods for solving hyperbo lic conservation laws\nby Hennes Hajduk (TU Dortmund) as part of CMO-B ound-Preserving Space and Time Discretizations\n\n\nAbstract\nDiscontinuou s Galerkin (DG) methods are among the most widely used numerical discretiz ation tech- niques for solving partial differential equations (PDEs). Thei r local conservation property\, inherent stability\, and\, favorable scala bility in parallel make these schemes attractive for many applications. Th ere are how- ever\, many shock-dominated examples\, for which even DG meth ods fail to produce stable approximations. To overcome this shortcoming\, we developed an algebraic flux limiter\, which blends a provably property- preserving low order method with a corresponding high order DG target sch eme. This monolithic convex limiter is primarily ]used to impose local (an d global) bounds on numerical approximations\, but extensions for incorpor ating entropy inequalities\, as well as relaxation of the constraints in s mooth regions are also possible. In my talk\, I will discuss the details o f the approach\, which include the sparsification of the low order method\ , stabilization of the numerical flux\, as well as the design of the monol ithic limiter. Sequential limiting for products of unknowns and the preser vation of global constraints\, such as nonnegativity of pres- sure will al so be touched upon and similarities to comparable schemes will be put into context. All presented numerical results were obtained with a code that i s based on the open source C++ library MFEM. The performance of the method will be evaluated by considering a variety of classical benchmarks for sc alar conservation laws\, as well as the systems of shallow water and Euler equations.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Yekaterina Epshteyn (University of Utah) DTSTART:20210824T160000Z DTEND:20210824T165000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/10 DESCRIPTION:Title: Numerical Methods for Shallow Water Models\nby Yekaterina Epshtey n (University of Utah) as part of CMO-Bound-Preserving Space and Time Disc retizations\n\n\nAbstract\nIn this talk\, we will discuss design of struct ure-preserving central-upwind finite volume methods for shallow water mode ls in domains with irregular geometry and for shallow water models with un certainty. Shallow water models are widely used in many scientific and eng ineering applications related to modeling of water flows in rivers\, lakes and coastal areas. Shallow water equations are examples of hyperbolic sys tems of balance laws and such mathematical models can present a significan t challenge for the construction of accurate and efficient numerical algor ithms.\nWe will show that the developed structure-preserving central-upwin d schemes for shallow water equations deliver high-resolution\, can handle complicated geometry\, and satisfy necessary stability conditions. We wil l illustrate the performance of the designed methods on a number of challe nging numerical tests. Current and future research will be discussed as we ll.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Arturo Vargas (Lawrence Livermore National Lab) DTSTART:20210824T182000Z DTEND:20210824T191000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/11 DESCRIPTION:Title: GPU Accelerated ALE Remap Strategies\nby Arturo Vargas (Lawrence Livermore National Lab) as part of CMO-Bound-Preserving Space and Time Dis cretizations\n\n\nAbstract\nIn this talk we present our work in updating t he high-order finite element-based ALE remap method in the MARBL multi-phy sics code from LLNL [1] for high performance on GPU platforms. MARBL is a multi- material hydrodynamics code based on a three phase Arbitrary-Lagran gian-Eulerian (ALE) framework: evolution of physical conservation laws wit hin a moving material (Lagrangian) frame\; mesh optimization\, and field r emap. The remap step corresponds to the transfer of state variables from t he initial mesh to the optimized mesh and can often dominate the run time for typical calculations. For high fidelity simulations it is imperative t hat the remap procedure be accurate\, preserve physical quantities\, and b e monotonic (not introduce new extrema). Furthermore\, for scalable perfor mance in large scale calculations it is imperative that the characteristic s of the algorithm are well suited for modern computing platforms (e.g. GP U based architectures).\nThis work is based on adopting a matrix-free algo rithmic approach. The current remap algorithm in MARBL is based on the wor k of Anderson and co-authors [1\, 2] which introduce a high-order approach based on concepts from flux corrected transport (FCT) and a discontinuous Galerkin (dG) discretization for the advection equation but requires full matrix assembly due to its algebraic nature. Methods based on full matrix assembly are known to have poor performance as the order of the method is increased. In addition\, there are involved memory motion operations whic h do not work well on GPU architectures. The new matrix-free framework we have been developing combines the residual distribution schemes of Hajduk and co-authors [3\,4] with a high-order DG scheme using the clip scale str ategy of Anderson et al. in [5]. Lastly\, we describe the algorithmic tail oring for the GPU and present performance comparisons between the differen t frameworks.\nThis work was performed under the auspices of the U.S. Depa rtment of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-ABS-824642.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Chris Kees (LSU College of Engineering) DTSTART:20210824T191000Z DTEND:20210824T200000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/12 DESCRIPTION:Title: Bound-preserving discretizations for variably saturated flow in porou s media\nby Chris Kees (LSU College of Engineering) as part of CMO-Bou nd-Preserving Space and Time Discretizations\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Volker John (Weierstrass Institute for Applied Analysis and Stocha stics) DTSTART:20210825T130000Z DTEND:20210825T135000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/13 DESCRIPTION:Title: Techniques for Improving Finite Element Solutions of Steady- State Co nvection-Diffusion Equations\nby Volker John (Weierstrass Institute fo r Applied Analysis and Stochastics) as part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstract\nIn the first part of the talk\, th e SUPG method for continuous piecewise linear finite elements is considere d. Numerical solutions computed with these methods are known to possess sp urious oscillations in a vicinity of layers. In [JKS11]\, a general approa ch for optimizing the stabilization parameter was presented. This talk wil l address an open problem stated in this paper: the restriction of the opt imization to subregions were the choice of the stabilization parameter is essential. In this way\, a reduction of the dimension of the space for opt imization is achieved. Suitable algorithms are discussed and numerical stu dies are presented. The second part of the talk deals with discontinuous G alerkin (DG) finite element methods. These methods are known to be stable and to compute sharp layers in the convection-dominated regime\, but also to show large spurious oscillations. Post-processing methods for reducing spurious oscillations are discussed\, which re- place the DG solution in a vicinity of layers by a constant or linear approximation. A survey of met hods that are available in the literature is presented and several general izations and modifications are proposed. Numerical studies illustrate the behavior of these methods. Details can be found in [FJ21]. This talk prese nts joint work with Ulrich Wilbrandt (WIAS) and Derk Frerichs (WIAS).\nRef erences\nFJ21 Derk Frerichs and Volker John. On reducing spurious oscillat ions in dis- continuous Galerkin (DG) methods for steady-state convection- diffusion equations. J. Comput. Appl. Math.\, 393:113487\, 20\, 2021.\nJKS 11 Volker John\, Petr Knobloch\, and Simona B. Savescu. A posteriori op- t imization of parameters in stabilized methods for convection-diffusion pro blems—Part I. Comput. Methods Appl. Mech. Engrg.\, 200(41-44):2916– 29 29\, 2011.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Abhinav Jha (RWTH Aachen University) DTSTART:20210825T135000Z DTEND:20210825T144000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/14 DESCRIPTION:Title: A Residual based a Posteriori Error Estimators for Algebraic Flux Cor rection Scheme\nby Abhinav Jha (RWTH Aachen University) as part of CMO -Bound-Preserving Space and Time Discretizations\n\n\nAbstract\nNon-linear discretizations are necessary for convection-diffusion-reaction equations for obtaining accurate solutions that satisfy the discrete maximum princi ple (DMP). Algebraic stabilizations\, also known as Algebraic Flux Correct ion (AFC) schemes\, belong to the very few finite element discretizations that satisfy this property. Results regarding the convergence of the schem e [1] and efficient solution of the nonlinear system of equations [2] have been obtained recently.. The talk is devoted to the proposal of a new res idual based a posteriori error estimator for AFC schemes. We derive a glob al upper bound in the energy norm of the system which is independent of th e choice of the limiter in the AFC scheme. We also derive a global upper b ound by combining the estimators from [3] and the AFC schemes. Numerical s imulations in 2d are presented which support the analytical findings.\nRef erences\n1. Gabriel R. Barrenechea\, Volker John\, and Petr Knobloch. Anal ysis of algebraic flux correction schemes. SIAM J. Numer. Anal.\, 54(4):24 27–2451\, 2016.\n2. Abhinav Jha and Volker John. A study of solvers for nonlinear AFC discretizations of convection- diffusion equations. Comput. Math. Appl.\, 78(9):3117–3138\, 2019.\n3. Volker John and Julia Novo. A robust SUPG norm a posteriori error estimator for stationary convection- d iffusion equations. Comput. Methods Appl. Mech. Engrg.\, 255:289–305\, 2 013.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Laura Saavedra Lago (Politecnica de Madrid) DTSTART:20210825T151000Z DTEND:20210825T160000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/15 DESCRIPTION:Title: Second order invariant domain preserving ALE approximation of Euler e quations\nby Laura Saavedra Lago (Politecnica de Madrid) as part of CM O-Bound-Preserving Space and Time Discretizations\n\n\nAbstract\nIn this t alk we will describe a second-order continuous finite element technique fo r solving hyperbolic systems in the arbitrary Lagrangian Eulerian framewor k (ALE). The main property of the method presented is that\, provided the user-defined ALE velocity is reasonable\, the approximate solution produce d by the algorithm is formally second-order accurate in space\, is conserv ative and preserves as many convex invariant sets of the hyperbolic system as desired by the user\, by using a convex limiting technique. The time s tepping is explicit\, the approximation in space is done with continuous f inite elements.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Jean-Luc Guermond (Texas A&M University) DTSTART:20210825T160000Z DTEND:20210825T165000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/16 DESCRIPTION:Title: Robust invariant domain preserving approximation of the compressible Navier-Stokes equations\nby Jean-Luc Guermond (Texas A&M University) a s part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstrac t\nThe objective of this talk is to present a fully-discrete approximation technique for the compressible Navier-Stokes equations. The method is imp licit-explicit\, second-order accurate in time and space\, and guaranteed to be invariant domain preserving. The restriction on the time-step size i s the standard hyperbolic CFL condition. One key originality of the method is that it is guaranteed to be conservative and invariant domain preservi ng under the standard hyperbolic CFL condition. The method is numerically illustrated on the OAT15a airfoil in the critical transonic regime at Re=3 millions. This is a joint work with M. Kronbichler\, M. Maier\, B. Popov and I. Tomas.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Tovar (Texas A&M University) DTSTART:20210825T182000Z DTEND:20210825T191000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/17 DESCRIPTION:Title: Hyperbolic Relaxation Technique For Solving The Dispersive Serre Equa tions\nby Eric Tovar (Texas A&M University) as part of CMO-Bound-Prese rving Space and Time Discretizations\n\n\nAbstract\nWe introduce a relaxat ion technique for solving the Serre Equations for dispersive water waves. The novelty of this technique is the reformulation of the Serre Equations into a hyperbolic system which allows for explicit time stepping in the nu merical method. We then propose a second-order approximation of the model using continuous finite elements that is well-balanced and positivity pres erving. The method is then numerically validated and illustrated by compar ison with laboratory experiments.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Hendrik Ranocha (University of Münster) DTSTART:20210826T130000Z DTEND:20210826T135000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/18 DESCRIPTION:Title: On stability of positivity-preserving Patankar-type time integration methods\nby Hendrik Ranocha (University of Münster) as part of CMO-Bo und-Preserving Space and Time Discretizations\n\n\nAbstract\nPatankar-type scheme are linearly implicit time integration methods constructed to sati sfy positivity properties of certain ordinary differential equations. Sinc e they are outside of the class of general linear methods\, they can have superior positivity preserving properties. However\, standard notions of s tability do not apply. For example\, classical linear stability analysis c annot be used since the schemes do not commute with diagonalization. Hence \, new concepts of stability need to be introduced.\nWe provide preliminar y investigations of stability of Patankar-type schemes. In particular\, we demon- strate problematic behavior of these methods that can lead to unde sired oscillations or order reduction. Extreme cases of the latter manifes t as spurious steady states. We investigate stability properties of vari- ous classes of Patankar-type schemes based on classical Runge-Kutta method s\, strong stability preserving Runge-Kutta methods\, and deferred correct ion schemes.\nThis project is joint work with Davide Torlo and Philipp O ̈ffner.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Philipp Offner (Johannes Gutenberg University Mainz) DTSTART:20210826T135000Z DTEND:20210826T144000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/19 DESCRIPTION:Title: Convergence of DG Schemes for the Euler Equations via Dissipative Wea k Solutions\nby Philipp Offner (Johannes Gutenberg University Mainz) a s part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstrac t\nSince the Cauchy problem for the complete Euler system is in general il l- posed in the class of admissible entropy weak solutions\, one searches alternatives and here the concept of dissipative weak solutions seems quit e promising to analyze this system analytically and numerically. In [1]\, the authors have studied the convergence properties of a class of entropy dissipative finite volume schemes for the barotropic and complete compress ible Euler equations in the multidimensional case and could prove suitable stability and consistency properties to ensure convergence of the FV sche mes via dissipative measure- valued solutions. In a series of paper\, the theory has been further developed for several (classical) FV schemes (of m aximum order two) and have been tested numerically\, cf. [2\,3]. In this t alk\, we consider as well convergence via dissipative weak solutions for t he Euler equation\, but focus on high-order finite element based methods\, in particular on a specific discontinuous Galerkin schemes. For the conve rgence proof\, we need certain properties like the preservation of several phyiscal quantities and some entropy estimates. We demonstrate how we ens ure these and prove convergence of our DG scheme via dissipative weak solu tions. In numerical simulations\, we verify our theoretical findings.\nRef erences\n1. E. Feireisl\, M. Luk ́aˇcov ́a-Medvid’ova ́ and H. Mizer ov ́a. Convergence of finite volume schemes for the Euler equations via d issipative measure-valued solutions\, Foundations of Computational Mathema tics\, 20 (4) pp. 1–44\, 2019.\n2. E. Feireisl\, M. Luk ́aˇcov ́a-Med vid’ova ́ and H. Mizerov ́a. A finite volume scheme for the Euler syst em inspired by the two velocities approach\, Numerische Mathematik\, 144(1 )\, pp. 89–132\, 2020.\n3. E. Feireisl\, M. Luk ́aˇcov ́a-Medvid’ov a ́\, B. She and Y. Wang. Computing oscillatory solutions of the Euler sy stem via K-convergence\, Mathematical Models and Methods in Applied Scienc es\, pp. 1–40\, 2021.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitri Kuzmin (TU Dortmund University) DTSTART:20210826T151000Z DTEND:20210826T160000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/20 DESCRIPTION:Title: Limiter-based entropy fixes for flux-corrected discretizations of non linear hyperbolic problems\nby Dmitri Kuzmin (TU Dortmund University) as part of CMO-Bound-Preserving Space and Time Discretizations\n\n\nAbstra ct\nWe consider continuous finite element approximations of hyperbolic pro blems and modify them to satisfy relevant inequality constraints. The prop osed approaches apply limiters to fluxes that represent the difference bet ween a high-order target scheme and a low-order property-preserving approx imation of Lax-Friedrichs type. In the first step of the limiting procedur e\, the given target fluxes are adjusted in a way that guarantees preserva tion of local and/or global bounds. In the second step\, additional limiti ng is performed\, if necessary\, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes impose entropy-conservative or entropy-dissipative bounds on entropy prod uction by antidiffusive fluxes and Runge-Kutta time discretizations. We pr esent three algorithms developed for this purpose. The semi- discrete (SD) fix is based on Tadmor’s entropy stability theory and constrains the sp atial semi-discretization. The fully discrete explicit (FDE) fix incorpora tes temporal entropy production into the flux constraints\, which makes th em more restrictive. The fully discrete implicit (FDI) fix performs iterat ive flux correction under SD-type constraints in the final Runge-Kutta sta ge. The effectiveness of these fixes is verified in numerical experiments for scalar equations and systems.\nReferences:\n1. D. Kuzmin and M. Quezad a de Luna\, Algebraic entropy fixes and convex limiting for continuous fin ite element discretizations of scalar hyperbolic conservation laws. Comput er Methods Appl. Mech. Engrg. 372 (2020) 113370.\n2. D. Kuzmin\, H. Hajduk and A. Rupp\, Limiter-based entropy stabilization of semi-discrete and fu lly discrete schemes for nonlinear hyperbolic problems. Preprint arXiv:210 7.11283 [math.NA]\, July 2021.\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Jesus Bonilla (LANL) DTSTART:20210826T160000Z DTEND:20210826T165000Z DTSTAMP:20260422T185218Z UID:CMO-21w5065/21 DESCRIPTION:Title: A Positivity-Preserving Finite Element Scheme for Keller-Segel Chemot axis Model\nby Jesus Bonilla (LANL) as part of CMO-Bound-Preserving Sp ace and Time Discretizations\n\n\nAbstract\nChemotaxis models describe the evolution of biological migration processes. In a migration process\, or- ganisms (or a group of cells) migrate in response to a chemical stimulus\ , which either attracts or repels them. In this work we focus on Keller-Se gel equations. A model that despite its biologically inaccurate results\, it is interesting and challenging from the mathematical point of view. Kel ler-Segel model solutions satisfy lower bounds\, and enjoy an energy law. The mathematical interest lies in developing numerical discretiza- tions t hat yield solutions that preserve these properties\, with the aim to contr ibute and give insights into discretizations for more complex and realisti c chemotaxis models. Positivity-preservation is very important in these mo dels to ensure physically meaningful results and proper evolution of the s olution. Otherwise\, it might lead to negative concentrations of cells or chemo-attractant. This is especially challenging in this model because\, d epending on the particular initial conditions specified\, it might lead to a blow-up of the solution. Recently\, Guti ́errez-Santacreu and Rodrigue z-Galv ́an [1] have published a numerical scheme for strictly acute meshe s\, that also yields solutions satisfying lower bounds and an energy law i n the discrete sense. In the present work\, we aim to extend these results to general meshes using an artificial diffusion stabilization method base d on [3]. Numerical results also show that the artificially added diffusio n does not significantly smear the solution\, while it enforces lower boun ds even for solutions that blow-up in a finite time.\nReferences\n1. J.V. Gutierr ́ez-Santacreu\, J.R. Rodriguez-Galv ́an\, Analysis of a fully di screte approximation for the classical Keller-Segel model: Lower and a pri ori bounds\, Comput. Math. Appl. 85 (2021) 69–81.\n2. S. Badia\, J. Boni lla\, J.V. Gutierr ́ez-Santacreu. Solving the Keller-Segel equations with finite element approximations over general meshes\, In preparation.\n3. B adia\, S.\, Bonilla\, J. Monotonicity-preserving finite element schemes ba sed on differentiable nonlinear stabilization\, Comput. Methods Appl. Mech . Engrg. 313 (2017) 133–15\n LOCATION:https://researchseminars.org/talk/CMO-21w5065/21/ END:VEVENT END:VCALENDAR