A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear advection-diffusion processes represented by (2+1)-dimensional Burgers equation. The IELDTM is adaptively constructed as stability preserved and high order time integrator for spatially discretized Burgers equation. For spatial discretization of the model equation, the Chebyshev spectral collocation method (ChCM) is utilized. A robust stability analysis and global error analysis of the IELDTM are presented with respect to the direction parameter θ. With the help of the global error analysis, adaptivity equations are derived to minimize the computational costs of the algorithms. The produced method is shown to eliminate the accuracy disadvantage of the classical θ-method and the stability disadvantages of the existing DTM-based methods. Two examples of the Burgers equation in one and two dimensions have been solved via the ChCM-IELDTM hybridization, and the produced results are compared with the literature. The present time integrator has been proven to produce more accurate numerical results than the MATLAB solvers, ode45 and ode15s.
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）
Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods
University of Pennsylvania（宾夕法尼亚大学）；Politecnico di Milano（米兰理工大学）；California Institute of Technology（加州理工学院）
Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs), but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-…
We propose a novel family of asymptotically stable, implicit-explicit, adaptive, time integration method (denoted with the $θ$-method) for the solution of the fractional advection-diffusion-reaction (FADR) equations. This family of time integration method generalized the computationally explicit $L_1$-method adopted by Brunner (J. Comput. Phys. {\bf 229} 6613-6622 (2010)) as well as the fully implicit method proposed by Jannelli (Comm. Nonlin. Sci. Num. Sim., {\bf 105}, 106073 (2022)). The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of …
We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Num…
In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integ…
In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz fractional advection-dispersion equation. This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are carried out to confirm the theoretical results of the proposed method.
We provide a preliminary comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity and the error in the phase speed of four spatiotemporal discretization schemes utilized for solving the one-dimensional (1D) linear advection diffusion reaction (ADR) equation: (a) An explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3) and the central difference scheme (CD2) for second order spatial discretization, (b) a fully implicit mid-point rule for time integration coupled with the OUCS3 and the Lele's compact s…
Time integration of advection dominated advection-diffusion problems on refined meshes can be a challenging task, since local refinement can lead to a severe time step restriction, whereas standard implicit time stepping is usually hardly suitable for treating advection terms. We show that exponential time integrators can be an efficient, yet conceptually simple, option in this case. Our comparison includes three exponential integrators and one conventional scheme, the two-stage Rosenbrock method ROS2 which has been a popular alternative to splitting methods for solving advection-diffusion pro…
We study the convergence of the discontinuous Galerkin (DG) method applied to the advection-reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order $\mathcal{O}(h^{p+1/2})$, which is the same rate as in the case of exact integration. Consequently, specialized quadrature rules that accurately integrate the non-smooth numerical fluxes are not required for …
In this paper, we develop a numerical resolution of the space-time fractional advection-dispersion equation. After time discretization, we utilize collocation technique and implement a product integration method in order to simplify the evaluation of the terms involving spatial fractional order derivatives. Then utilizing Bernstein polynomials as basis, the problem is transformed into a linear system of algebraic equations. Error analysis and order of convergence for the proposed method are also discussed. Some numerical experiments are presented to demonstrate the effectiveness of the propose…
We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improve…
We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter $δ$, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure f…
Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods
Department of Mathematics, The Florida State University（佛罗里达州立大学数学系）；ETSIAE-UPM - Universidad Politécnica de Madrid. School of Aeronautics（马德里理工大学航空工程学院）
Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent PDEs in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, …
This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the a…
A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge-Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compressio…
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical in…
Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models
BMW Group, Research and Innovation Centre（宝马集团研究与创新中心）；Dynamore GmbH（Dynamore有限公司）；Queen Mary, University of London（伦敦大学女王玛丽学院）
This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalizing previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge-Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang and Jameson [1] are performed to qua…
The monodomain model is widely used in in-silico cardiology to describe excitation propagation in the myocardium. Frequently, operator splitting is used to decouple the stiff reaction term and the diffusion term in the monodomain model so that they can be solved separately. Commonly, the diffusion term is solved implicitly with a large time step while the reaction term is solved by using an explicit method with adaptive time stepping. In this work, we propose a fully explicit method for the solution of the decoupled monodomain model. In contrast to semi-implicit methods, fully explicit methods…
The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy la…
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are positive, and the mass fraction of each species is between 0 and 1. Secondly, due to the rapid reaction rate, the system may contain stiff sources, and the strong-stability-preserving explicit Runge-Kutta method may result in limited time step sizes. To obtain physically relevant numerical approximations, we apply the bound-preserving technique to the DG methods. F…
We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability…
Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both space and time. This paper derives arbitrary order finite element exterior calculus spatial discretizations for the two-dimensional (2D) Navier-Stokes and drift-reduced magnetohydrodynamic equations that conserve both energy and enstrophy to machine precision when coupled with generally symplectic time-integration methods. Both continuous and discontinuous-Galerkin (DG) weak formulations can ensure …
