Linear Advection-diffusion Problem Research Articles

On the Stability of IMEX Upwind gSBP Schemes for 1D Linear Advection-Diffusion Equations

AbstractA fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts (upwind gSBP) schemes in space and implicit-explicit Runge-Kutta (IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly, while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first- and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin (DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay (BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space, since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.

Adomian Decomposition Method for Solving Boussinesq Equations Using Maple

This paper uses the Adomian Decomposition Method (ADM) to solve Boussinesq equations using Maple. The Boussinesq approximation for water waves is a weakly nonlinear and long-wave approximation in fluid dynamics. The approximation is named after Joseph Boussinesq, who developed it in response to John Scott Russell’s observation of a wave of translation (also known as solitary wave or soliton). Bossinesq’s article from 1872 introduced the equations that are now known as the Boussinesq equations. Numerical methods are commonly utilized to solve nonlinear equation systems. In this paper, we investigate a nonlinear singly perturbed advection-diffusion problem. Using the usual Adomian Decomposition Method, we formulate an approximate linear advection-diffusion problem and investigate several practical numerical approaches for solving it (ADM). The Adomian Decomposition Method (ADM) is a powerful tool for numerical simulations and approximation analytic solutions. The Adomian Decomposition Method (ADM) is used to solve nonlinear advection differential equations using Maple by illustrating numerous examples. The findings are presented in the form of tables and graphs for several examples. For various examples, the findings are presented in the form of tables and graphs. The difference between the precise and numerical solutions indicates the Maple program solution’s efficacy, as well as the ease and speed with which it was acquired.

The Spectral Difference Raviart–Thomas Method for Two and Three-Dimensional Elements and Its Connection with the Flux Reconstruction Formulation

The purpose of this work is to describe in detail the development of the spectral difference Raviart–Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the flux reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows building a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough analysis of the numerical dissipation and dispersion of both aforementioned schemes and the nodal discontinuous Galerkin FR (FR-DG) method with two and three-dimensional elements is proposed through the use of the combined-mode Fourier approach. SDRT is shown to possess an enhanced temporal linear stability in comparison to FR-DG. On the contrary, SDRT displays larger dissipation and dispersion errors with respect to FR-DG. Finally, the study is concluded with a set of numerical experiments, the linear advection-diffusion problem, the Isentropic Euler Vortex, and the Taylor-Green Vortex (TGV). The latter test case shows that SDRT schemes present a non-linear unstable behavior with simplex elements and certain polynomial degrees. For the sake of completeness, the matrix form of the SDRT method is developed and the computational performance of SDRT with respect to FR schemes is evaluated using GPU architectures.

Spatial eigenanalysis of spectral/hp continuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows

This study considers the spatial eigensolution analysis of spectral/hp continuous Galerkin (CG) schemes, complementing a recent work by Moura et al. (2016) [15] which addressed CG's temporal analysis. While the latter assumes periodic boundary conditions, the spatial approach presumes inflow/outflow type conditions and therefore provides insights for a different class of problems. The linear advection-diffusion problem is considered for a wide range of Péclet numbers, allowing for viscous effects at different intensities. The inviscid (linear advection) case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous Galerkin (DG) schemes in the limit of strong over-upwinding. These effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under-resolved simulations at high Reynolds numbers. The spectral vanishing viscosity (SVV) technique is subsequently considered as a natural stabilization strategy, in the context of linear advection. An optimization procedure is employed to match SVV diffusion levels to those of DG at appropriate polynomial orders. The resulting CG-SVV discretisations are tested against under-resolved computations of spatially developing vortex-dominated flows and display excellent robustness at high Reynolds numbers along with superior eddy-resolving characteristics at higher polynomial orders. This highlights the importance of appropriate stabilization techniques to improve the potential of spectral/hp CG methods for high-fidelity simulations of transitional and turbulent flows, including implicit LES / under-resolved DNS approaches.

L2-stability analysis of IMEX-(σ,μ)DG schemes for linear advection-diffusion equations

A fully discrete L2-stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the (σ,μ)-family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L2-stability for time steps Δt=O(d/a2), where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent (14,94)-recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stability property which is also confirmed by numerical experiments.

Stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and Bassi and Rebay II numerical fluxes for linear triangular elements

The flux reconstruction (FR) method has gained popularity within the research community. The approach has been demonstrated to recover high-order methods such as the discontinuous Galerkin (DG) method. Stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction (ESFR) methods also named Vincent–Castonguay–Jameson–Huynh (VCJH) methods. ESFR schemes can be viewed as DG schemes with modally filtered correction fields. Using this class of methods, the linear advection diffusion problem has been shown to be stable using the local discontinuous Galerkin scheme (LDG) to compute the viscous numerical flux. This stability proof has been extended for linear triangular and tetrahedra elements. Although the LDG scheme is commonly used, it requires, on particular meshes, a wide stencil, which raises the computational cost.As a consequence, many prefer the compact interior penalty (IP) or the Bassi and Rebay II (BR2) numerical fluxes. This article, for the first time, derives, for both schemes, a condition on the penalty term to ensure the stability for the linear diffusion equation. The derivations are based on Gauss Legendre flux points and on periodic boundary conditions. Moreover the article proposes that for both the IP and BR2 numerical fluxes, the stability of the ESFR scheme is independent of the auxiliary correction field. A von Neumann analysis is conducted to study the maximal time step of various ESFR methods.

On series solution for second order semilinear parabolic IBVPs

This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order ( N ), the number of eigenfunctions ( M ) and the diffusion coefficient is rather complex.

A Fully Conservative Eulerian-Lagrangian Stream-Tube Method for Advection-Diffusion Problems

We present a new method for a two-dimensional linear advection-diffusion problem of a “tracer” within an ambient fluid. The problem should have isolated external sources and sinks, and the bulk fluid flow is assumed to be governed by an elliptic problem approximated by a standard locally conservative scheme. The new method, the fully conservative Eulerian-Lagrangian stream-tube method, combines the volume corrected characteristics-mixed method with the use of a stream-tube mesh. Advection of the tracer is approximated using characteristic tracing in time of regions of space, which maintains mass conservation. However, the shape of a characteristic trace-back region is numerically approximated, so its volume must also be correct to maintain accurate approximation of the tracer density (i.e., the mass of the ambient fluid must be conserved during the advection step). Our new method has the advantages that it is fully locally conservative (both tracer and ambient fluid mass is conserved locally), has low numerical diffusion overall and no numerical cross-diffusion between stream-tubes, can use very large time steps (perhaps 20 to 30 times the CFL limited step), and can use a very coarse mesh, since it is tailored to the flow pattern. Because advection is approximated within stream-tubes, it is essentially one-dimensional, making it relatively easy to implement and computationally efficient. We also present a grid transfer technique to approximate more simply the physical diffusion on a rectangular grid rather than on the stream-tube mesh. The new method can be used for many applications, but especially problems of flow and transport in porous media, which have sources and sinks isolated to wells. Examples include the modeling of groundwater contaminant migration, petroleum production, and carbon sequestration.

A Note on Homogenization of Advection-Diffusion Problems with Large Expected Drift

This contribution is concerned homogenization of linear advection-diffusion problems with rapidly oscillating coefficient functions and large expected drift. Even though the homogenization of this type of problems is generally well known, there are several details that have not yet been treated explicitly or even not been treated at all. Here, we will have a special look at uniqueness, regularity, boundedness and equivalent formulations of the homogenized equation. In particular, we generalize results of Allaire and Raphael \[C. R. Math. Acad. Sci. Paris 344 (2007)(8), 523–528] and Donato and Piatnitski \[Multi Scale Problems and Asymptotic Analysis. Tokyo: Gakkotosho 2006, pp. 153–165]. The results obtained in this contribution are of special interest for the numerical analysis of multi-scale schemes to approximate the analytic solutions.

The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift

This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.

Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems

In this paper, we extend a wavelet-Taylor Galerkin method for linear and non-linear wave problems introduced in [B.V. Rathish Kumar, Mani Mehra, Wavelet-Taylor Galerkin method for Burger’s equations, BIT Numer. Math. 45 (3) (2005) 543–560, B.V. Rathish Kumar, Mani Mehra, Time accurate solution of Korteweg–de Vries equation using wavelet Galerkin method, Appl. Math. Comput. 162 (1) (2005) 447–460], where the authors have taken the advantage of the good approximation properties of the wavelets but not the useful feature of wavelets to compress the solution with a localized structure, which might occur intermittently anywhere in the computational domain. Here this method takes the advantage of the wavelet bases capabilities to compress operators and sparse representation of solutions with a localized structure, which occur in wave problems. We illustrate the performance of the method through linear advection–diffusion problem and non-linear waves problems taken from shallow water theory and non-linear optics.

A robust a posteriori estimator for the Residual-free Bubbles method applied to advection-diffusion problems

We develop the a posteriori error analysis for the RFB method, applied to the linear advection-diffusion problem: the numerical error, measured in suitable norms, is estimated in terms of the numerical residual. The robustness is investiged, in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error.

Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations

Simple, mesh/grid free, explicit and implicit numerical schemes for the solution of linear advection-diffusion problems is developed and validated herein. Unlike the mesh or grid-based methods, these schemes use well distributed quasi-random points and approximate the solution using global radial basis functions. The schemes can be seen as generalized finite differences with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no need for complex mesh/grid structure and with no extra implementation difficulties.

An Exponentially Fitted Finite Volume Method for the Numerical Solution of 2D Unsteady Incompressible Flow Problems

In this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the Navier-Stokes equations describing 2D incompressible flows. The method is based on an unstructured Delaunay mesh and its dual Dirichlet tessellation, combined with a locally constant approximation to the flux. This yields a piecewise exponential approximation to the exact solution. Numerical tests are presented for a linear advection-diffusion problem with boundary layers. The method is then applied to the driven cavity problem with Reynolds numbers up to 10 4. The numerical results indicate that the method is robust for a wide range of values of the Reynolds number. In the case Re = 10 4 unsteady solutions are captured if the mesh is sufficiently fine.