Das Buch besteht aus Beiträgen von Plenarrednern und anderen Rednern auf der ICOSAHOM 2023, die vom 14. bis 18. August 2023 an der Yonsei University in Korea stattfand. Die Themen des Buches konzentrieren sich auf die jüngsten Fortschritte bei der Analyse und Anwendung spektraler und hochrangiger Methoden. Insbesondere enthält sie mehrere Umfragepapiere, die einen Überblick über die jüngsten Fortschritte und Beschreibungen zukünftiger Richtungen in den verwandten Fachgebieten bieten. Das Buch wird für Wissenschaftler und Ingenieure interessant sein, die sich für numerische Analysen und wissenschaftliche Berechnungen interessieren, insbesondere für spektrale und hochentwickelte Methoden.
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Über dieses Buch
The book consists of contributions by plenary speakers and other speakers at the ICOSAHOM 2023 which was held at Yonsei University, Korea during Aug 14-18, 2023 . The subjects of the book focus on recent advances on the analysis and applications of spectral and high-order methods. In particular, it includes several survey papers which provide reviews of recent progresses and descriptions of future directions in the related subjects. The book will be of interest to scientists and engineers who are interested in numerical analysis and scientific computing, particularly spectral and high-order methods.
In this paper, we study the two dimensional (2D) rotating nonlinear Schrödinger equation with the focusing nonlinearity, modeling the attractive Bose-Einstein condensates (BECs) under a rotating frame. For the 2D case, by employing the Fourier pseudo-spectral method/finite difference methods/time splitting methods, extensive numerical investigations are devoted to the existence/nonexistence of vortices in ground states, dynamics of the center-of-mass, resonance-type phenomenon, vortex stability and blow-up. From our numerical results, we find that there exists no vortex in the ground states of 2D attractive rotating BECs with harmonic traps. Then we focus on the dynamical behavior of the focusing rotating nonlinear Schrödinger equations, which share similar properties to the defocusing case. Finally, we examine numerically the sharp criteria for the blow-up analysis. Extensive numerical tests and theoretical justification are provided.
Mimetic methods construct discrete numerical schemes based on discrete analogs of spatial differential vector calculus operators like divergence, gradient, curl, Laplacian, etc. They mimic solution symmetries, conservation laws, vector calculus identities, and other important properties of continuum partial differential equations models. The original versions of these methods were restricted to be of low-order of accuracy. High-order mimetic operators were later introduced, first by Castillo and Grone at San Diego State University, via the introduction of convenient inner product weights to enforce a discrete high-order extended Gauss divergence theorem, and later by a collaboration of Los Alamos National Laboratory and a group of researchers at Milano-Pavia. This review focuses on the developments of high-order mimetic differences by Castillo and his group at San Diego and the utilization of these techniques to different applications. In addition, when appropriate, it exhibits similarities and differences between the two methodologies.
An edgewise iterative scheme for high order discontinuous Galerkin method in time combined with discontinuous Galerkin methods with Lagrange multiplier in space (DG-DGLM) is developed for the large system resulting from the discretization of the nonlinear hyperbolic scalar conservation laws. Convergence analysis of the iterative scheme is given for the linear problem. Newton iteration is applied elementwise over the product elements of space and time for the nonlinear problem. Several numerical examples are presented.
This paper gives a survey of our recent work on spacetime spectral methods for PDEs and an exposition of some current progress. Classical spectral methods for time-dependent PDEs use low-order finite difference discretization of the time derivative, and spectral discretization of the spatial derivatives, creating a large imbalance in the temporal and spatial discretization errors. Space-time spectral methods address this deficiency by employing spectral discretization in time as well. Two main advantages of space-time spectral methods include full spectral convergence and ease of implementation for PDEs defined on regular geometry. The method is extremely robust across all types of PDEs (dispersive, diffusive, presence of advection terms with all standard boundary conditions). The main drawback of space-time spectral methods is that time marching is no longer feasible – all unknowns in both space and time must be solved for simultaneously.
Avleen Kaur, Shaun Lui, Sarah Nataj, Chandramali Piyasundara Wilegoda Liyanage
We first briefly review some recently proven new results about \(Q^k\) spectral element method for second order linear PDEs, including its order of accuracy as a finite difference method in \(\ell ^2\)-norm and monotonicity, both of which are special properties of \(Q^k\) spectral element method on structured meshes. We discuss some extensions or applications of these two special properties, including the accuracy for the Helmholtz equation and applications of monotone discrete Laplacian to a semi-linear problem. In particular, the \(Q^2\) spectral element method gives a fourth order accurate monotone discrete Laplacian, with which one can obtain explicit convergence rates of Picard and Newton iterations for solving a special second order semilinear PDE.
Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We perform a cell average decomposition of the LWFR scheme that is similar to the one used in the admissibility preserving framework of Zhang and Shu (J Comput Phys 229(9):3091–312, 2010). By performing a flux limiting of the time averaged numerical flux, the decomposition is used to obtain an admissibility preserving LWFR scheme. The admissibility preservation framework is further extended to a newly proposed extension of the LWFR scheme for conservation laws with source terms. This is the first extension of the high order LW scheme that can handle source terms. The admissibility and accuracy are verified by numerical experiments on the Ten Moment equations of Livermore et al.
This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) scheme to solve second order time-dependent partial differential equations in a conservative form on curvilinear meshes. The method uses the BR1 scheme to reduce the system to first order so that the LWFR scheme can be applied. The work makes use of embedded error-based time stepping recently introduced for LWFR schemes which becomes particularly relevant in the absence of a CFL stability limit for parabolic equations. The scheme is verified to show optimal order of convergence and validated with transonic flow over airfoil and unsteady flow over cylinder.
Finite Element Methods are increasingly used in various scientific and industrial applications due to their advantageous combination of high-order accuracy, geometric flexibility, and robustness. However, their computational cost increases rapidly when the solution is discretised with higher-order polynomial approximations. For this reason, much research has been devoted to overcome this drawback. This work presents the implementation of an improved p-MultiGrid algorithm based on the nonlinear Full Approximation Scheme in a discontinuous Galerkin solver for the solution of the three-dimensional and compressible Navier-Stokes and Reynolds-Average Navier-Stokes equations for flows in both laminar and turbulent conditions. Only the application of these algorithms, in both the h and p forms, to the solution of the Euler and Navier-Stokes equations for laminar flows is well documented in the literature.
Daniel Bulgarini, Antonio Ghidoni, Gianmaria Noventa
Computational fluid dynamics (CFD) is becoming increasingly important relevant in many industrial and scientific fields, promoting the development of more accurate and efficient CFD solvers. In particular, a modern CFD solver should be characterized by (i) a direct interface with geometric modelling software to reduce the generation time of the computational grid, (ii) the ability to use different approaches for the accurate description of turbulent flows, and (iii) a dynamic adaptation algorithm of the computational mesh to reduce computing time. Isogeometric high-order discontinuous Galerkin (dG) methods, where B-splines or non-uniform rational B-splines (NURBS) are used to describe both geometry and unknowns, can serve as a viable option for achieving these objectives, because they guarantee an exact representation of the geometry and high accuracy, even when using unstructured, strongly distorted, and curved meshes. In the present work, an isogeometric high-order dG solver is extended to solve the Reynolds Averaged Navier-Stokes (RANS) equations and Spalart-Allmaras turbulence model equations for compressible flows, am it is validated in the computation of some benchmark test-cases.
Daniel Bulgarini, Antonio Ghidoni, Gianmaria Noventa, Stefano Rebay
In this chapter, we present a high-order numerical methodology based on Discontinuous Galerkin schemes for the accurate simulation of shock-turbulence interactions. The method combines a positivity-preserving limiter that guarantees the robustness of simulations, an artificial diffusion operator that suppresses shock oscillations, and a vortex sensor that preserves the artificial diffusion to impact turbulent eddies. The ability of the method to represent accurately turbulence with a robust shock capture is demonstrated from simulations of the compressible Taylor-Green vortex at Mach 1.25 and Reynolds 1600.
We propose a block finite difference, error inhibiting scheme that is fourth-order accurate for short to moderate times and has a sixth-order convergence rate for long times. This scheme outperforms the standard fourth-order Finite Difference scheme. We also demonstrate that the proposed scheme is a particular type of nodal-based Discontinuous Galerkin method with \(p=1\).
A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey’s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers’ equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.
Wai Sun Don, Jia-Le Li, Leevan Ling, Bao-Shan Wang, Yinghua Wang
The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schrödinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in \(H^s\) with \(0<s\le 2\). Here, this analysis is extended to dimensions \(d=3, 4, 5\) for data satisfying \(d/2-1<s\le 2\). In this setting, convergence of order s/2 in \(L^2\) is proven. Numerical examples illustrate these convergence results.
The iterative solution of spectral/hp element methods for complex geometries is challenging due to the typically large problem sizes, various length scales involved, anisotropic spatial discretisation and denser linear systems compared to the lower-order counterparts. These factors lead to ill-conditioned matrices, and efficient preconditioning techniques are needed to reduce the computational cost of these methods, facilitating their adoption in industrial applications. The current work compares the design and performance of various preconditioners within the incompressible Navier-Stokes equations solver of the open-source spectral/hp element method framework Nektar++. A new preconditioner is proposed within Nektar++, Lower-Order Refined (LOR) preconditioner, constructed on a low-order (P\(=\) 1) finite element discretisation spectrally equivalent to a given high-order discretisation. The LOR preconditioner provides advantages like cheap operator evaluations, constant memory requirement per degree of freedom, minimal sensitivity to high aspect-ratio elements and bounded iterative condition number with increasing problem size under certain conditions. The performance of the proposed preconditioner is compared to legacy implementations of preconditioners within Nektar++ and other existing out-of-the-box algebraic multigrid (AMG) methods to determine the best preconditioner for the current application. The chosen test cases are relevant in the context of race-car aerodynamics.
Parv Khurana, Spencer J. Sherwin, Julien Hoessler, Francesco Bottone, David Moxey
Simulating industrial flows involving complex geometries with high-order CFD methods requires good-quality curvilinear meshes to ensure their accuracy, stability and computational efficiency. We describe a modification of the high-order mesh generator NekMesh to allow the curving of straight-sided meshes generated by third-party software to benefit from their robustness and flexibility. We propose further a modified workflow that maintains mesh conformity to the underlying CAD (B-rep) model, but most importantly it allows us to assess the geometrical accuracy and its potential impact on the flow simulation. Application to a simple benchmark test and a complex automotive geometry will demonstrate the suitability of the modified workflow and the resulting mesh quality.
Kaloyan S. Kirilov, Joaquim Peiró, Jingtian Zhou, Mashy D. Green, David Moxey
We develop a numerical scheme for solving the advection equation of \(\mathbb {S}^2\)-valued functions of real variables, which models the time-evolution of a \(\mathbb {S}^2\)-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of \(\mathbb {S}^2\). To handle \(\mathbb {S}^2\)-functions which have kinks or sharp discontinuity in their components, we incorporate the Spherical Essentially Non-Oscillatory (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of \(\mathbb {S}^2\)-functions.
In this work, we introduce a novel application of the adaptive mesh refinement (AMR) technique for the global stability analysis of incompressible flows. The design of an accurate mesh for transitional flows is crucial since an inadequate resolution might introduce numerical noise that triggers premature transition to turbulence. With AMR, we enable the design of three different and independent meshes for the non-linear base flow and the linear direct and adjoint solutions. Each of these is designed to reduce the truncation and quadrature errors for its respective solution, based on the spectral error indicator. We provide details about the workflow and the refining procedure. The numerical framework is validated for the two-dimensional flow past a circular cylinder, computing a portion of the spectrum for the linearised direct and adjoint Navier–Stokes operators.
Daniele Massaro, Valerio Lupi, Adam Peplinski, Philipp Schlatter
We present an hr-adaptivity framework for morphing a given mesh to fit a target surface prescribed as the zero isocontour of a discrete function. In this framework, high-order meshing is posed as a variational minimization problem that depends on the mesh quality prescribed via the target matrix optimization paradigm (TMOP) and position of a subset of mesh nodes with respect to the target surface. The proposed formulation ensures that the variational problem is converged and mesh quality degradation near the surface is limited, even when the mesh topology is incompatible with the target surface. Additionally, a mesh subset-based approach and h-refinement is introduced to efficiently increase fitting accuracy while reducing the computational cost of the mesh morphing problem. The hr-adaptivity technique extends to different element types in two- and three-dimensions, and can be used in existing finite element and spectral element frameworks to obtain high-order body-fitted meshes. Various numerical experiments demonstrate the robustness and accuracy of the fitting approach for problems of practical interest such as Lagrangian hydrodynamics and topology optimization.
Ketan Mittal, Jorge-Luis Barrera, Tzanio Kolev, Mathias Schmidt, Vladimir Tomov
Hamiltonian systems are known to conserve the Hamiltonian function, which describes the energy evolution over time. Obtaining a numerical spatio-temporal scheme that accurately preserves the discretized Hamiltonian function is often a challenge. In this paper, the use of high order mimetic difference methods is investigated for the numerical solution of Hamiltonian equations. The mimetic operators are based on developing high order discrete analogs of the vector calculus quantities divergence and gradient. The resulting high order operators preserve the properties of their continuum ones, and are therefore said to mimic properties of conservation laws and symmetries. Symplectic fourth order schemes are implemented in this paper for the time integration of Hamiltonian systems. A theoretical framework for the energy preserving nature of the resulting schemes is also presented, followed by numerical examples.
The pursuit of more precise numerical methods in Computational Fluid Dynamics (CFD) is crucial, especially given the inherently nonlinear and highly complex nature of the governing equations for physical phenomena. Therefore, the present work proposes an extension of a computational methodology based on the coupling of the Fourier Pseudospectral Method (FPSM) and the Immersed Boundary Method (IBM), applied to simulations of external flows over aerodynamic profiles. This hybrid approach, termed IMERSPEC, capitalizes on the inherent advantages of pseudospectral methods, of fering high accuracy and low computational costs, facilitated by the Fast Fourier Transform algorithm. IBM is applied to enforce non-periodic boundary conditions on the Navier-Stokes equations, as FPSM requires periodicity at the boundaries to achieve convergence. The aerodynamic behavior of the analyzed profiles is investigated by determining lift and drag coefficients as functions of the angle of attack, with results validated against existing literature. The obtained results demonstrate consistency with literature findings, particularly in flows at lower Reynolds numbers, underscoring the efficacy of the IMERSPEC methodology and the employed physical models in simulating complex flows.
Laura Augusta Vasconcelos de Albuquerque, Mariana Fernandes dos Santos Villela, Felipe Pamplona Mariano
In this paper, based on the bifurcation theory, we numerically calculate and visualize non-trivial multiple solutions of the singularly perturbed Neumann problem on a square using the Jacobi pseudospectral method. Starting from the non-zero solution of the corresponding problem, we numerically obtain multiple positive solutions with various symmetries by the branch switching method. The bifurcation diagrams are drawn, displaying the symmetry breaking bifurcation phenomena of the singularly perturbed Neumann problem. Numerical results demonstrate the effectiveness of this method.
The spectral/hp element method provides an efficient approach for implicit Large-Eddy Simulations (iLES) of incompressible flows. However, the simulation of highly-resolved incompressible flow on industrial geometries is often limited by computational resources. This is particularly severe in advection-dominated flows where high Reynolds numbers cause strong stability restrictions on the maximum-allowable time step size and, thus, significantly increase the time-to-solution. This issue renders algorithms like velocity-correction Schemes inefficient due to their explicit treatment of the advection terms. In an effort to reduce computational costs for industrial geometries, we investigate a velocity-correction scheme with linearly-implicit treatment of the advection terms based on the work of [6, 19]. We consider an extension of this work to determine the possible efficiency gain in quasi-3D simulations where one resolves the three-dimensional nature of the flow whilst reducing computational cost through the assumption of a homogeneous/spectral dimension. This algorithm gains efficiency by treating only the mean-mode of each velocity component implicitly while treating higher-frequency modes explicitly. Thus, the algorithm stabilises the computation by making the bulk of the energy implicit. Initial results show that the scheme is capable of a strong improvement in stability. The present paper present our progress along with numerical tests.
Henrik Wüstenberg, Spencer J. Sherwin, Joaquim Peiró, David Moxey
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