ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
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, 07:07:39pm CET

Session Overview 
Session  
Z6: RBF, meshless, unfitted
 
Presentations  
12:00pm  12:20pm
Oversampled and collocation RBFFD methods for solving nonlinear conservation laws Uppsala University, Department of Information Technology, Sweden We present our latest results on solving nonlinear conservation laws using radial basis function generated finite difference (RBFFD) method. The analytic solutions to nonlinear conservation laws are known to develop shocks (discontinuities). Shocks lead to an occurrence of nonphysical oscillations that pollute the numerical solutions (Gibbs phenomenon), and can cause spurious growth of the numerical solution in time. To stabilize the shocks when using the RBFFD method, we apply a residual based viscosity (RV) filter borrowed from the finite element community. Here the residual of the conservation law gives an approximate location of a shock, and by that enables a selective smearing of the solution. We focus on applying RV to two settings of the RBFFD method: (i) a collocation setting, (ii) an oversampled setting. A set of numerical experiments in two dimensions show that the solution in both settings converges with highorder when RV is active and the solution is smooth. The experiments also show that RV is an effective stabilization of the RBFFD method when solving Burger’s equation and the KurganovPetrovaPopov rotating wave problem. 12:20pm  12:40pm
The leastsquares RBFFD method ^{1}Uppsala University, Sweden; ^{2}University of Massachusetts Dartmouth, MA, USA The radial basis functiongenerated finite difference (RBFFD) method is based on stencil approximations over scattered nodes. This provides the simplicity of finite difference methods, combined with flexibility with respect to the geometry. RBFFD methods work well in practice, but it has been hard to provide formal convergence proofs. There is also some sensitivity to node layouts, especially in the vicinity of boundaries with conditions including derivatives. Here, we introduce a leastsquares formulation of the method where each stencil is used for several evaluation points such that the overall linear system of equations to solve becomes overdetermined. This improves the stability of the approximations and the performance of the method. Furthermore, it allows us to prove convergence by considering a nearby continuous leastsquares problem. We show that highorder convergence results can be achieved for Poisson test problems with mixed boundary conditions in two and three dimensions. 12:40pm  1:00pm
A highorder unfitted RBFFD method for solving stationary PDEs ^{1}Uppsala University, Department of Information Technology, Sweden; ^{2}University of Massachusetts Dartmouth, Department of Mathematics, USA Radial basis function generated finite difference (RBFFD) methods for PDEs require a set of interpolation points which conform to a computational domain. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of the computational domain. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of the computational domain. In this talk we present a modification to an RBFFD method which allows the interpolation points to be placed in a box that encapsulates the computational domain. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving PDEs with mixed boundary conditions show that our method is robust and highly accurate. Compared with the conventional approaches, the approximation error tends to be smaller when stencil sizes are large. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm. 1:00pm  1:20pm
A high order meshless finite difference method for elliptic interface problems University of Giessen, Germany Elliptic interface problems are solved by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the meshbased finite difference or finite element methods with the same order. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to 6 for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately. Results are obtained jointly with Mansour Safarpoor. Full article is published by Applied Numerical Mathematics, online version: https://doi.org/10.1016/j.apnum.2020.11.018 1:20pm  1:40pm
Exploring the mortar and pointtopoint interpolation methods for handling nonconformal interfaces in highorder discontinuous Galerkin methods University of Exeter, United Kingdom The capability to incorporate moving geometric features within models for complex simulations is a common requirement in many fields, for example rotational and sliding components in the modelling of turbomachinery. The inclusion of this movement results in the introduction of nonconformal interfaces separating the physical mesh along the regions of movement by detaching nodal connections. In this talk we will consider the numerical effects of two methods used to carry information across this interface in the setting of highorder discontinuous Galerkin methods. The first is the socalled mortar approach, where flux integrals along edges are split according to the positioning of the nonconformal grid. The second is a lessdocumented pointtopoint interpolation method, where the interior and exterior quantities for flux evaluations are interpolated from elements lying on the opposing side of the interface. Although the mortar approach has significant advantages in terms of its numerical properties, in that it preserves the local conservation properties of DG methods, in the context of complex 3D meshes it poses notable implementation difficulties which the pointtopoint method handles more readily. We explore the numerical properties of each method, focusing not only on observing convergence orders for smooth solutions, but also how each method performs in underresolved simulations of linear and nonlinear hyperbolic problems, to inform the use of these methods in implicit largeeddy simulations. 1:40pm  2:00pm
Development and application of an immersed boundary method for highorder flux reconstruction schemes ^{1}ETSIAEUPMSchool of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E28040 Madrid, Spain; ^{2}NUMECA International S.A., Chaussee de la Hulpe 187, Brussels, B1170, Belgium; ^{3}Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain Highorder methods for computational fluid dynamics are known to provide improved accuracy with relatively lower cost over loworder schemes, particularly for unsteady flows, which have the potential to form the next generation of industrial software [1]. However, the use of these methods on unstructured grids requires the generation of curved grids, which can be challenging for complex configurations. Immersed boundary method (IBM) delivers an alternative solution to avoid the complexity of mesh generation through handling nonconforming grids and moving boundaries in Cartesian domains [2]. With the capability of local hprefinement to reduce the computational cost, developing IBM in the context of highorder framework becomes a promising area of research. In this work, we introduce our development in combining IBM with highorder flux reconstruction schemes [3,4]. Volume penalization [5,6] is chosen to handle the IBM treatment due to its good convergence property, robustness and ease of implementation. Theoretical study based on von Neumann analysis, numerical experiments for static and moving geometries [7], as well as applications to fluidstructure interaction are reported. References: [1] Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T. and Kroll, N., 2013. High‐order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 72(8), pp.811845. [2] Mittal, R. and Iaccarino, G., 2005. Immersed boundary methods. Annu. Rev. Fluid Mech., 37, pp.239261. [3] Huynh, H.T., 2007, June. A flux reconstruction approach to highorder schemes including discontinuous Galerkin methods. In 18th AIAA Computational Fluid Dynamics Conference (p. 4079). [4] Vincent, P.E., Castonguay, P. and Jameson, A., 2011. A new class of highorder energy stable flux reconstruction schemes. Journal of Scientific Computing, 47(1), pp.5072. [5] Angot, P., Bruneau, C.H. and Fabrie, P., 1999. A penalization method to take into account obstacles in incompressible viscous flows. Numerische Mathematik, 81(4), pp.497520. [6] Kolomenskiy, D. and Schneider, K., 2009. A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. Journal of Computational Physics, 228(16), pp.56875709. [7] Kou, J., Joshi, S., HurtadodeMendoza, A., Puri, K., Hirsch, C., and Ferrer, E., 2021. Immersed boundary method for highorder flux reconstruction based on volume penalization. http://doi.org/10.5281/zenodo.4437325 
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