We propose a new family of high-order explicit generalized-$α$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$ matrix systems explicitly and updating the other $2k$ variables at each time-step. The user can control the numerical dissipation in the discrete spectrum's high-frequency regions by adjusting the method's coefficients. We study the method's spectrum behaviour and show that the CFL condition is independent of the accuracy order. The stability region remains invariant while we increase the accuracy order. Next, we exploit efficient preconditioners for the isogeometric matrix to minimize the computational cost. These preconditioners use a diagonal-scaled Kronecker product of univariate parametric mass matrices; they have a robust performance with respect to the spline degree and the mesh size, and their decomposition structure implies that their application is faster than a matrix-vector product involving the fully-assembled mass matrix. Our high-order schemes require simple modifications of the available implementations of the generalized-$α$ method. Finally, we present numerical examples demonstrating the methodology's performance regarding single- and multi-patch IGA discretizations.
Isogeometric spectral approximation for elliptic differential operators
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷大学计算研究所）；Department of Applied Geology, Curtin University（科廷大学应用地质系）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织矿产资源部）
Dispersion-minimized mass for isogeometric analysis
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷计算研究所，科廷大学）；Department of Applied Geology, Western Australian School of Mines, Curtin University（西澳矿业学院应用地质系，科廷大学）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织（CSIRO）矿产资源部）
Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
Johannes Kepler University（约翰内斯·开普勒大学）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）
Higher order accurate mass lumping for explicit isogeometric methods based on approximate dual basis functions
Technical University of Darmstadt（达姆施塔特工业大学）；Otto von Guericke University Magdeburg（奥托·冯·吉尔克马格德堡大学）；Brandenburg University of Technology Cottbus-Senftenberg（勃兰登堡科技大学科特布斯-森夫滕贝格分校）
Modal Integrators for Isogeometric Stabilization of Outliers
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin（德克萨斯大学奥斯汀分校航空航天工程与应用力学系）
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context
