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.
In this article, we address the solution of advection-dominated flow problems by stabilised methods, by means of least-squares computed stabilised coefficients. As main methodological tool, we introduce a data-driven off-line/on-line strategy to compute them with low computational cost. We compare the errors provided by the least-squares stabilised coefficients to those provided by several previously established stabilised coefficients within the solution of advection-diffusion and Navier-Stokes flows, on structured and un-structured grids, with and Lagrange Finite Elements up to third degree …
Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection-dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation, which enables stabilization through an enrichment process. The presented work uses solution-tailored enrichments for the numerical solution of the one-dimensional, unsteady Burgers equation. Mainly, generalizable exponential and hyperbolic tangent enrichments effectively capture local, steep boundary layer/shock features. Results show natural alleviation of osc…
In this work, we investigate (energy) stability of global radial basis function (RBF) methods for linear advection problems. Classically, boundary conditions (BC) are enforced strongly in RBF methods. By now it is well-known that this can lead to stability problems, however. Here, we follow a different path and propose two novel RBF approaches which are based on a weak enforcement of BCs. By using the concept of flux reconstruction and simultaneous approximation terms (SATs), respectively, we are able to prove that both new RBF schemes are strongly (energy) stable. Numerical results in one and…
We propose using machine learning and artificial neural networks (ANNs) to enhance residual-based stabilization methods for advection-dominated differential problems. Specifically, in the context of the finite element method, we consider the streamline upwind Petrov-Galerkin (SUPG) stabilization method and we employ ANNs to optimally choose the stabilization parameter on which the method relies. We generate our dataset by solving optimization problems to find the optimal stabilization parameters that minimize the distances among the numerical and the exact solutions for different data of diffe…
Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov $N$--width of these problems. In the last years, new nonlinear algorithms obtained smaller reduced spaces. In these works only the offline phase of these algorithms was shown. In this work, we study MOR algorithms for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. We propose an arbit…
Motivated by challenges in Earth mantle convection, we present a massively parallel implementation of an Eulerian-Lagrangian method for the advection-diffusion equation in the advection-dominated regime. The advection term is treated by a particle-based, characteristics method coupled to a block-structured finite-element framework. Its numerical and computational performance is evaluated in multiple, two- and three-dimensional benchmarks, including curved geometries, discontinuous solutions, pure advection, and it is applied to a coupled non-linear system modeling buoyancy-driven convection in…
We propose a combination of the Eulerian Lagrangian Localised Adjoint Method (ELLAM) and the Modified Method of Characteristics (MMOC) for time-dependent advection-domina\-ted PDEs. The combined scheme, so-called GEM scheme, takes advantages of both ELLAM scheme (mass conservation) and MMOC scheme (easier computations), while at the same time avoids their disadvantages (respectively, harder tracking around the injection regions, and loss of mass). We present a precise analysis of mass conservation properties for these three schemes, and after achieving global mass balance, an adjustment yieldi…
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…
We present stability and error analysis for algebraic flux correction schemes based on monolithic convex limiting. For a continuous finite element discretization of the time-dependent advection equation, we prove global-in-time existence and the worst-case convergence rate of 1/2 w.r.t. the L2 error of the spatial semi-discretization. Moreover, we address the important issue of stabilization for raw antidiffusive fluxes. Our a priori error analysis reveals that their limited counterparts should satisfy a generalized coercivity condition. We introduce a limiter for enforcing this condition in t…
Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are two complementary singular-value decomposition (SVD) techniques that are widely used to construct reduced-order models (ROMs) in a variety of fields of science and engineering. Despite their popularity, both DMD and POD struggle to formulate accurate ROMs for advection-dominated problems because of the nature of SVD-based methods. We investigate this shortcoming of conventional POD and DMD methods formulated within the Eulerian framework. Then we propose a Lagrangian-based DMD method to overcome this so-called trans…
Adaptive hyperviscosity stabilisation for the RBF-FD method in solving advection-dominated transport equations
Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Jozef Stefan International Postgraduate School（约瑟夫·斯特凡国际研究生院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）
Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs
Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism
Middle East Technical University（中东理工大学）；Institute of Applied Mathematics, Middle East Technical University（中东理工大学应用数学研究所）；Department of Mathematics, Middle East Technical University（中东理工大学数学系）
This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these …
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, …
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 introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the assocociated difficulties is a semi-Lagrangian based reconstruction of subgrid variablity into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a…
