Most Download

Published in last 1 year | In last 2 years| In last 3 years| All| Most Downloaded in Recent Month | Most Downloaded in Recent Year|

In last 2 years

Please wait a minute...


For Selected: Download Citations EndNote Ris BibTeX Toggle Thumbnails

Select

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refnement Will Pazner, Tzanio Kolev Communications on Applied Mathematics and Computation    2022, 4 (2): 697-727.   DOI: 10.1007/s42967-021-00136-3 Abstract （1499）      PDF       Knowledge map    Save In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with hp-refnement. These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refned technique, which in this work, we extend to the hp-refnement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh h, and the degree of the polynomial approximation p. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refned meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2). Reference | Related Articles | Metrics | Comments（0）

Select

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang Communications on Applied Mathematics and Computation    2022, 4 (2): 477-499.   DOI: 10.1007/s42967-021-00128-3 Abstract （1350）      PDF       Knowledge map    Save In Chen et al. (J. Sci. Comput. 81(3):2188-2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defned on simplicial meshes, for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k ≥ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3):635-650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defned on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k ≥ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory. Reference | Related Articles | Metrics | Comments（0）

Select

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations Lu Zhang, Daniel Appelö, Thomas Hagstrom Communications on Applied Mathematics and Computation    2022, 4 (3): 855-879.   DOI: 10.1007/s42967-021-00149-y Abstract （1706）      PDF       Knowledge map    Save We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant. Reference | Related Articles | Metrics | Comments（0）

Select

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields Dinshaw S. Balsara, Roger Käppeli Communications on Applied Mathematics and Computation    2022, 4 (3): 945-985.   DOI: 10.1007/s42967-021-00166-x Abstract （1667）      PDF       Knowledge map    Save This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger's equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PNPM-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times. Reference | Related Articles | Metrics | Comments（0）

Select

Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System Jiawei Sun, Shusen Xie, Yulong Xing Communications on Applied Mathematics and Computation    2022, 4 (2): 381-416.   DOI: 10.1007/s42967-021-00119-4 Abstract （2154）      PDF       Knowledge map    Save Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, the BBM-BBM system, the Bona-Smith system, etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well. Reference | Related Articles | Metrics | Comments（0）

Select

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation Lorenzo Botti, Daniele A. Di Pietro Communications on Applied Mathematics and Computation    2022, 4 (3): 783-822.   DOI: 10.1007/s42967-021-00142-5 Abstract （1769）      PDF       Knowledge map    Save We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $ L^2 $-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes. Reference | Related Articles | Metrics | Comments（0）

Select

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods Xiaozhou Li Communications on Applied Mathematics and Computation    2022, 4 (3): 759-782.   DOI: 10.1007/s42967-021-00144-3 Abstract （1760）      PDF       Knowledge map    Save Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order $ k+1 $ to order $2k+1$ in the $ L^2 $ norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters. Reference | Related Articles | Metrics | Comments（0）

Select

Discontinuous Galerkin Method for Macroscopic Traffic Flow Models on Networks Lukáš Vacek, Václav Kučera Communications on Applied Mathematics and Computation    2022, 4 (3): 986-1010.   DOI: 10.1007/s42967-021-00169-8 Abstract （1689）      PDF       Knowledge map    Save In this paper, we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads. On individual roads, we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters. To solve traffic flows on networks, we construct suitable numerical fluxes at junctions based on preferences of the drivers. We prove basic properties of the constructed numerical flux and the resulting scheme and present numerical experiments, including a junction with complicated traffic light patterns with multiple phases. Differences with the approach to numerical fluxes at junctions from Čanić et al. (J Sci Comput 63: 233–255, 2015) are discussed and demonstrated numerically on a simple network. Reference | Related Articles | Metrics | Comments（0）

Select

Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids Mahboub Baccouch Communications on Applied Mathematics and Computation    2022, 4 (2): 437-476.   DOI: 10.1007/s42967-021-00123-8 Abstract （1349）      PDF       Knowledge map    Save This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form -Δu=f (x, y, u) on Cartesian grids. By introducing special GaussRadau projections and using duality arguments, we obtain, under some suitable choice of numerical fuxes, the optimal convergence order in L2-norm of O(hp+1) for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order p + 2 toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves (p + 1)-th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results. Reference | Related Articles | Metrics | Comments（0）

Select

On Periodic Oscillation and Its Period of a Circadian Rhythm Model Miao Feng, Chen Zhang Communications on Applied Mathematics and Computation    2022, 4 (3): 1131-1157.   DOI: 10.1007/s42967-021-00146-1 Abstract （1776）      PDF       Knowledge map    Save We theoretically study periodic oscillation and its period of a circadian rhythm model of Neurospora and provide the conditions for the existence of such a periodic oscillation by the theory of competitive dynamical systems. To present the exact expression of the unique equilibrium in terms of parameters of system, we divide them into eleven classes for the Hill coefficient $ n=1 $ or $ n=2 $, among seven classes of which nontrivial periodic oscillations exist. Numerical simulations are made among the seven classes and the models with the Hill coefficient $ n=3 $ or $ n=4 $ to reveal the influence of parameter variation on periodic oscillations and their periods. The results show that their periods of the periodic oscillations are approximately 21.5 h, which coincides with the known experiment result observed in constant darkness. Reference | Related Articles | Metrics | Comments（0）

Select

Discontinuous Galerkin Methods for a Class of Nonvariational Problems Andreas Dedner, Tristan Pryer Communications on Applied Mathematics and Computation    2022, 4 (2): 634-656.   DOI: 10.1007/s42967-021-00133-6 Abstract （1370）      PDF       Knowledge map    Save We extend the fnite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2):786-801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the "fnite element Hessian" as an auxiliary variable in the formulation. Representing the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al. (SIAM J. Numer. Anal. 39(5):1749-1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the fnite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting. Reference | Related Articles | Metrics | Comments（0）

Select

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations Ohannes A. Karakashian, Michael M. Wise Communications on Applied Mathematics and Computation    2022, 4 (3): 823-854.   DOI: 10.1007/s42967-021-00143-4 Abstract （1734）      PDF       Knowledge map    Save The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type, coupled through their nonlinear terms. In our previous work [9], we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes. In this sequel, we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in [9]. The key tool employed to effect our analysis is the dispersive reconstruction developed by Karakashian and Makridakis [20] for related discontinuous Galerkin methods. We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators. Reference | Related Articles | Metrics | Comments（0）

Select

Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence Mohammed Homod Hashim, Akil J. Harfash Communications on Applied Mathematics and Computation    2022, 4 (3): 1011-1056.   DOI: 10.1007/s42967-021-00124-7 Abstract （1710）      PDF       Knowledge map    Save In this paper, a finite element scheme for the attraction-repulsion chemotaxis model is analyzed. We introduce a regularized problem of the truncated system. Then we obtain some a priori estimates of the regularized functions, independent of the regularization parameter, via deriving a well-defined entropy inequality of the regularized problem. Also, we propose a practical fully discrete finite element approximation of the regularized problem. Next, we use a fixed point theorem to show the existence of the approximate solutions. Moreover, a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived. In addition, the uniqueness of the fully discrete approximations is preformed. Finally, we discuss the convergence to the fully discrete problem. Reference | Related Articles | Metrics | Comments（0）

Select

Development of a Balanced Adaptive Time-Stepping Strategy Based on an Implicit JFNK-DG Compressible Flow Solver Yu Pan, Zhen-Guo Yan, Joaquim Peiró, Spencer J. Sherwin Communications on Applied Mathematics and Computation    2022, 4 (2): 728-757.   DOI: 10.1007/s42967-021-00138-1 Abstract （1468）      PDF       Knowledge map    Save A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations. A proper relation between the spatial, temporal and iterative errors generated within one time step is constructed. With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization, an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization. The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection, steady-state fow past a fat plate, Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3 900. The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady, well-resolved and under-resolved simulations. Reference | Related Articles | Metrics | Comments（0）

Select

Parametric Regression Approach for Gompertz Survival Times with Competing Risks H. Rehman, N. Chandra Communications on Applied Mathematics and Computation    2022, 4 (4): 1175-1190.   DOI: 10.1007/s42967-021-00154-1 Abstract （3179）      PDF       Knowledge map    Save Regression models play a vital role in the study of data regarding survival of subjects. The Cox proportional hazards model for regression analysis has been frequently used in survival modelling. In survival studies, it is also possible that survival time may occur with multiple occurrences of event or competing risks. The situation of competing risks arises when there are more than one mutually exclusive causes of death (or failure) for the person (or subject). In this paper, we developed a parametric regression model using Gompertz distribution via the Cox proportional hazards model with competing risks. We discussed point and interval estimation of unknown parameters and cumulative cause-specific hazard function with maximum-likelihood method and Bayesian method of estimation. The Bayes estimates are obtained based on non-informative priors and symmetric as well as asymmetric loss functions. To observe the finite sample behaviour of the proposed model under both estimation procedures, we carried out a Monte Carlo simulation analysis. To demonstrate our methodology, we also included real data analysis. Reference | Related Articles | Metrics | Comments（0）

Select

Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula Ren-jun Qi, Zhi-zhong Sun Communications on Applied Mathematics and Computation    2022, 4 (4): 1313-1350.   DOI: 10.1007/s42967-021-00177-8 Abstract （3018）      PDF       Knowledge map    Save With the help of the asymptotic expansion for the classic L1 formula and based on the L1- type compact difference scheme, we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation. Three extrapolation formulas are presented, whose temporal convergence orders in L∞-norm are proved to be 2, 3-α, and 4-2α, respectively, where 0 < α < 1. Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal 4-γ and 6-2γ order in L∞-norm, respectively, where 1 < γ < 2. Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation, the fast extrapolation methods are obtained which reduce the computational complexity significantly while keeping the accuracy. Several numerical experiments confirm the theoretical results. Reference | Related Articles | Metrics | Comments（0）

Select

A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks Rami Masri, Charles Puelz, Beatrice Riviere Communications on Applied Mathematics and Computation    2022, 4 (2): 500-529.   DOI: 10.1007/s42967-021-00126-5 Abstract （1379）      PDF       Knowledge map    Save This paper formulates an efficient numerical method for solving the convection diffusion solute transport equations coupled to blood flow equations in vessel networks. The reduced coupled model describes the variations of vessel cross-sectional area, radially averaged blood momentum and solute concentration in large vessel networks. For the discretization of the reduced transport equation, we combine an interior penalty discontinuous Galerkin method in space with a novel locally implicit time stepping scheme. The stability and the convergence are proved. Numerical results show the impact of the choice for the steady-state axial velocity profile on the numerical solutions in a fifty-five vessel network with physiological boundary data. Reference | Related Articles | Metrics | Comments（0）

Select

Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations Xiaoying Han, Habib N. Najm Communications on Applied Mathematics and Computation    2022, 4 (4): 1457-1493.   DOI: 10.1007/s42967-022-00188-z Abstract （3332）      PDF       Knowledge map    Save Numerical algorithms for stiff stochastic differential equations are developed using linear approximations of the fast diffusion processes, under the assumption of decoupling between fast and slow processes. Three numerical schemes are proposed, all of which are based on the linearized formulation albeit with different degrees of approximation. The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems. Convergence analysis is conducted for one of the schemes, that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1. Approximations arriving at the other two schemes are discussed. Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems. Reference | Related Articles | Metrics | Comments（0）

Select

Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods Andreas Dedner, Robert Klöfkorn Communications on Applied Mathematics and Computation    2022, 4 (2): 657-696.   DOI: 10.1007/s42967-021-00134-5 Abstract （1537）      PDF       Knowledge map    Save This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial diferential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly fexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs. Reference | Related Articles | Metrics | Comments（0）

Select

Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods Jennifer K. Ryan Communications on Applied Mathematics and Computation    2022, 4 (2): 417-436.   DOI: 10.1007/s42967-021-00121-w Abstract （1316）      PDF       Knowledge map    Save This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifcally, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) fltering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of diferent resolutions. Although this article presents the details of the SIAC flter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data. Reference | Related Articles | Metrics | Comments（0）