我们考虑一个既非协调又对流占优的对流扩散方程。我们提出一种可能的数值方法，据我们所知是新的，基于原始方程相关的不变测度。该方法在[C. Le Bris, F. Legoll和F. Madiot, C. R. Acad. Sci. Paris, Serie I, vol. 354, 799-803 (2016)]中被总结。我们证明该方法允许无条件良的有限元近似。我们提供了数值分析和一组全面的数值测试，显示该方法可以稳定、准确且比传统稳定化方法更稳健。
英文摘要：
We consider an advection-diffusion equation that is both non-coercive and advection-dominated. We present a possible numerical approach, to our best knowledge new, and based on the invariant measure associated to the original equation. The approach has been summarized in [C. Le Bris, F. Legoll and F. Madiot, C. R. Acad. Sci. Paris, Serie I, vol. 354, 799-803 (2016)]. We show that the approach allows for an unconditionally well-posed finite element approximation. We provide a numerical analysis and a set of comprehensive numerical tests showing that the approach can be stable, as accurate as, and more robust than a classical stabilization approach.
We devise a stabilized method to weakly enforce bound constraints in the discrete solution of advection-dominated diffusion problems. This method combines a nonlinear penalty formulation with a discontinuous Galerkin-based residual minimization method. We illustrate the efficiency of this scheme for both uniform and adaptive meshes through proper numerical examples.
We devise and analyze a class of the primal discontinuous Galerkin methods for the magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we find a new mechanism under the vector case that provides more flexibility in constructing the schemes. For the more general Friedrichs system, we show the stability and optimal error estimate, which boil down to two core ingredients -- the weight function and the special projection -- that contain information of advection. Numerical experiments are provided to verify the theoretical results.
In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the sol…
In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framew…
In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov-Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.
We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme…
We construct, analyse and assess various schemes of second order of accuracy in space and time for model advection-diffusion-reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial problems and are derived following two related approaches, namely ADER and MUSCL-Hancock. Detailed analysis of linear stability and local truncation error are carried out. In addition, the schemes are implemented and assessed for various test problems. Empirical convergence rate studies confirm the theoretically expected accuracy in both space and time.
We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed appr…
In this study, an implicit-explicit local differential transform method (IELDTM) based on Taylor series representations is produced for solving 2D and 3D advection-diffusion equations. The parabolic advection-diffusion equations are reduced to the nonhomogeneous elliptic system of partial differential equations with the utilization of the Chebyshev spectral collocation approach in temporal variable. The IELDTM is constructed over 2D and 3D meshes using continuity equations of the neighbour representations with either explicit or implicit forms in related directions. The IELDTM is proven to hav…
A stochastic method is described for estimating Green's functions (GF's), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF's, within any geometry, the technique solves the central challenge in obtaining Green's function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF's that can be used: a) to obtain (infinite) sets of solutions,…
We study the computation of coupled advection-diffusion-reaction equations by the Schwarz waveform relaxation method. The study starts with linear equations, and it analyzes the convergence of the computation with a Dirichlet condition, a Robin condition, and a combination of them as the transmission conditions. Then, an optimized algorithm for the Dirichlet condition is presented to accelerate the convergence, and numerical examples show a substantial speedup in the convergence. Furthermore, the optimized algorithm is extended to the computation of nonlinear equations, including the viscous B…
In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilize the advantages of each of these methods. The usage of HMM allows us to handle anisotropic diffusion tensors on generic polygonal (polytopal) grids; whereas the CF method provides a framework for the construction of a uniformly seco…
The aim of this study is to compare numerical methods for the simulation of single-phase flow and transport in fractured media, described here by means of the Discrete Fracture Network (DFN) model. A Darcy problem is solved to compute the advective field, then used in a subsequent time dependent transport-diffusion-reaction problem. The numerical schemes are benchmarked in terms of flexibility in handling geometrical complexity, mass conservation and stability issues for advection dominated flow regimes. To this end, three benchmark cases have been specifically designed and are here proposed, …
Numerical solutions of a 2D fluid problem coupled to a nonlinear non-local reaction-advection-diffusion problem for cell crawling migration in a discoidal domain
Université Paul Sabatier（图卢兹第三大学）；Université Paris Descartes（巴黎笛卡尔大学）
Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations
本文在 DeC 框架下构造隐式与隐式–显式 ADER、DeC 方法，通过 Runge–Kutta 重解释和 von Neumann 分析刻画其 ODE 及线性 PDE 稳定性，并给出系数边界与格式选择建议。
In this work, we consider an advection-diffusion equation, coupled to a Poisson equation for the velocity field. This type of coupling is typically encountered in models arising from plasma physics or porous media flow. The aim of this work is to build upon the complete flux scheme (an improvement over the Scharfetter-Gummel scheme by considering the contribution of the source term), so that its second-order convergence, which is uniform in Péclet numbers, carries over to these models. This is done by considering a piecewise linear approximation of the velocity field, which is then used for de…
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…
We are interested in the long-time behaviour of approximate solutions to heterogeneous and anisotropic linear advection-diffusion equations in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. We consider two linear methods, as well as a new, nonlinear scheme, for which we prove the existence and the positivity of discrete solutions. We show that the discrete solutions to the three schemes converge exponentially fast in time towards the associated discrete steady-states. To illustrate our theoretical findings, we present some numerical simulations asse…
In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is $(4-α)$, where $α~(0<α<1)$ is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of generalized fractional advection-diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss about the stability and convergence of the difference scheme. Numerical examples are prese…
This paper develops a physics-informed neural network (PINN) based on the Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the formulation by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural n…
The balancing domain decomposition methods (BDDC) are originally introduced for symmetric positive definite systems and have been extended to the nonsymmetric positive definite system from the linear finite element discretization of advection-diffusion equations. In this paper, the convergence of the GMRES method is analyzed for the BDDC preconditioned linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. Compared to the finite element discretizations, several additional norms for the numerical trace have to be used and the equivale…
In the present study, an advection-diffusion problem has been considered for the numerical solution. The continuum equation is discretized using both upwind and centered scheme. The linear system is solved using the ILU preconditioned BiCGSTAB method. Both Dirichlet and Neumann boundary condition has been considered. The obtained results have been compared for different cases.
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 present convergence analysis of operator learning in [Chen and Chen 1995] and [Lu et al. 2020], where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection-diffusion equations with or without reaction. We find that the convergence rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.
We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds $\mathbb{M} \subset \mathbb{R}^3$ of co-dimension one. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently-developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis perfo…
