Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints
Institute for Problems in Mechanics, Russian Academy of Sciences（俄罗斯科学院力学问题研究所）；University of Sonora（索诺拉大学）；Bauman Moscow State Technical University（鲍曼莫斯科国立技术大学）；National Research Nuclear University MEPhI（国立核研究医科大学）
Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions are unknown a priori (the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative. The second method is based on introducing a new auxiliary non-local variable with the subsequent transformation to the Cauchy problem for the corresponding system of ODEs. The third method is based on adding to the original equation of a differential constraint, which is an auxiliary ODE connecting the given variables and a new variable. The proposed methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the transformed problems admit the application of standard fixed-step numerical methods. The efficiency of these methods is illustrated by solving a number of test problems that admit an exact analytical solution. It is shown that: (i) the methods based on non-local transformations of a special kind are more efficient than several other methods, (ii) among the proposed methods, the most general method is the method based on the differential constraints. Some examples of nonclassical blow-up problems are considered. Simple theoretical estimates are derived for the critical value of an independent variable. It is shown that the method based on a non-local transformation of the general form as well as the method based on the differential constraints admit generalizations to the $n$th-order ODEs and systems of coupled ODEs.
Institute for Problems in Mechanics, Russian Academy of Sciences（俄罗斯科学院力学问题研究所）；Bauman Moscow State Technical University（鲍曼莫斯科国立技术大学）；National Research Nuclear University MEPhI（国立研究核子大学MEPhI）；University of Sonora（索诺拉大学）
Based on trace theory, we study efficient methods for concurrent integration of B-spline basis functions in IGA-FEM. We consider several scenarios of parallelization for two standard integration methods; the classical one and sum factorization. We aim to efficiently utilize hybrid memory machines, such as modern clusters, by focusing on the non-obvious layer of the shared memory part of concurrency. We estimate the performance of computations on a GPU and provide a strategy for performing such computations in practical implementations.
This paper aims to develop numerical approximations of the Keller--Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localiz…
We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We v…
This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best cand…
Meshfree GMsFEM-based exponential integration for multiscale 3D advection-diffusion problems
North-Eastern Federal University（东北联邦大学）；Wenzhou-Kean University（温州肯恩大学）；Khalifa University of Science and Technology（哈利法科学技术大学）；Universidad Nacional Autónoma de Honduras（洪都拉斯国立自治大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）
This paper presents two new non-classical Lagrange basis functions which are based on the new Jacobi-Müntz functions presented by the authors recently. These basis functions are, in fact, generalizations form of the newly generated Jacobi based functions. With respect to these non-classical Lagrange basis functions, two non-classical interpolants are introduced and their error bounds are proved in detail. The pseudo-spectral differentiation (and integration) matrices have been extracted in two different manners. Some numerical experiments are provided to show the efficiency and capability of t…
The Finite Cell Method (FCM) together with Isogeometric analysis (IGA) has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid-structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, b…
We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary Method for spatial discretization is combined with an explicit predictor/multi-corrector time integration to integrate in time the numerical solutions to the shallow water equations, both for the full and reduced-order model. In order to improve the approximation of the solution manifold also for geometries that are untested during the offline stage, the snapshot…
We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generati…
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced ba…
Reduced Basis A Posteriori Error Bounds for the Instationary Stokes Equations
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University（亚琛计算工程科学高等研究院）；Faculty of Civil Engineering, RWTH Aachen University（亚琛工业大学土木工程学院）
