Linear Advection-diffusion Equation Research Articles

Multi-USV cooperative oil spill source seeking via extremum seeking combined with information fusion

This paper addresses the issue of cooperative source seeking for oil spill pollution by multiple unmanned surface vehicles (USVs), where the oil spill plume propagation is modeled by a linear advection–diffusion equation in two-dimensional space. The main objective is to construct a cooperative source seeking scheme for a group of USVs in the framework of extremum seeking combined with information fusion to solve such cooperative source seeking problem. To do this, both a gradient-based control law and a formation control law are included in the cooperative source seeking scheme. The gradient-based control law that is constructed by extremum-seeking control combined with information fusion drives the multiple USVs to approach the oil spill source, where the gradient is estimated online via the extremum-seeking control and the fused concentration value that is derived via a linear H∞ filter from the local concentration measurements at the USV’s current position. In the source seeking process, the USVs driven by the formation control law based on the virtual structure method maintain a preset circular formation. The convergence performance of the cooperative source seeking scheme is analyzed rigorously by the Lyapunov method. Finally, extensive numerical simulations are included to validate the theoretical algorithm.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

Fractional advection-diffusion equations have demonstrated to be a powerful tool in modeling complex anomalous diffusion in applied science. In this paper, we studied novel linear time-fractional advection-diffusion equations associated with an extension of Mittag-Leffler fractional derivative operator. A useful feature of the used extension is to address the limitations of the Mittag-Leffler fractional derivative model. We, mainly, proposed a numerical approach to provide approximate solutions to linear time-fractional advection-diffusion equations with the studied extended fractional derivative operator. The suggested approach is based on discretizing the studied models with respect to spatio-temporal domain using uniform meshes. A new type of solutions for the studied models was generated numerically using the proposed approach. Besides, a comparative study was conducted to verify the accuracy and feasibility of the proposed approach.

Solutions of fuzzy advection-diffusion and heat equations by natural adomian decomposition method

In this article, we present an algorithm for computing analytical solutions of linear fuzzy advection-diffusion equations and one-dimensional fuzzy heat equations involving an external source. The fuzzy problems can be solved by using the natural transform and Adomian decomposition method. The results obtained through the natural Adomian decomposition method are calculated in a series form that converges rapidly to the exact solution. To enhance the practicality of our work, we provide examples to illustrate our findings.

A numerical approach to the optimal control of thermally convective flows

The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the L2-projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiments.

A modified equation analysis for immersed boundary methods based on volume penalization: Applications to linear advection–diffusion equations and high-order discontinuous Galerkin schemes

The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection–diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immersed boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection–diffusion equations.

Eigenanalysis and non-modal analysis of collocated discontinuous Galerkin discretizations with the summation-by-parts property

Guided by the von Neumann and non-modal analyses, we investigate the dispersion and diffusion properties of collocated discontinuous Galerkin methods with the summation-by-parts property coupled with the simultaneous approximation technique. We use the linear advection and linear advection-diffusion equations as model problems. The analysis is carried out by varying the order of the spatial discretization, the Péclet number, and looking at the effect of the upwind term. The eigenanalysis is verified to provide insights into the numerical errors based on the behavior of the primary mode. The dispersion and diffusion errors associated with the spatial discretization are shown to behave according to the primary- or physical- mode in the range of low wavenumbers. In this context, the discretization of the model problems is verified to be stable for all flow regimes and independent of the solution polynomial degree. The effect of the upwind term shows that its effect decreases by increasing the accuracy of the discretization. Further, two analyses that includes all modes are used to get better insights into the diffusion and robustness of the scheme. From the non-modal analysis, the short-term diffusion is computed for different flow regimes and solution polynomial degrees. The energy decay or long-term diffusion based on all eigenmodes based on the eigenmodes matrix is analyzed for t > 0 and different spatial discretizations. The results are validated against under-resolved turbulence simulations of the Taylor–Green vortex at two Reynolds numbers, i.e. , Re = 100 and 1600, and a Mach number Ma = 0.1 , and the decaying of compressible homogeneous isotropic turbulence at a Reynolds number based on the Taylor microscale of Re λ = 100 and a turbulent Mach number of Ma t = 0.6 .

Long-time behaviour of hybrid finite volume schemes for advection–diffusion equations: linear and nonlinear approaches

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 assessing long-time behaviour and positivity. We also compare the accuracy of the schemes on some numerical tests in the stationary case.

Stable timestep formulas for high-order advection–diffusion and Navier–Stokes solvers

We derive three-dimensional stable timestep formulas for high-order explicit time integration of the advection–diffusion equation. The proposed formulas cover explicit first, second, third, and fourth-order Runge–Kutta integrators in time as well as upwind, central, second-order, high-order upwind (k-schemes), and flux-limiters for the advection term along with central discretization on the diffusion term. The stability rules are then compared to numerical computations of stable timesteps of the advection–diffusion equation and show excellent agreement. Finally, the formulas are implemented in a finite volume Navier–Stokes code and used to guide stable timestep selection to assess their suitability for Navier–Stokes solvers. All cases that were tested showed stable non-diverging solutions and significant improvement over common CFL-type stability specification. Despite its basis in the linear advection–diffusion equation, this work could be useful in providing a starting point for stable timestep selection for nonlinear evolutionary-type PDEs such as the Navier–Stokes equations.

A brief review on the solutions of advection-diffusion equation

In this work both linear and nonlinear advection-diffusion equations are considered and discussed their analytical solutions with different initial and boundary conditions. The work of Ogata and Banks, Harleman and Rumer, Cleary and Adrian, Atul Kumar et al., Mojtabi and Deville are reviewed for linear advection-diffusion equations and for nonlinear, we have chosen the work of Sakai and Kimura. Some enthusiastic functions used in the articles, drawbacks and applications of the results are discussed. Reduction of the advection-diffusion equations into diffusion equations make the governing equation solvable by using integral transform method for analytical solution. For nonlinear advection-diffusion equations, the Cole-Hopf transformation is used to reduce into the diffusion equation. Different dispersion phenomena in atmosphere, surface and subsurface area are outlined.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Two-stage explicit schemes based on numerical approximations of convection-diffusion equations

A wide family of finite difference methods for the convection-diffusion problems based on an explicit two-stage scheme and a seemingly implicit method are presented. In this paper, to have a greater stability region while maintaining the second-order accuracy, a family of methods that combines the MacCormack and Saul'vey schemes is proposed. The stability analysis of the combined methods is investigated using the von Neumann approach. In each case, it is found that it is the convection term that limits the stability of the scheme. Based on the von Neumann analysis, valuable stability limits in terms of mesh parameters for maintaining accurate results are determined in an analytic manner and demonstrated through computer simulations. Two model problems consisting of linear advection-diffusion and nonlinear viscous Burgers equation are given to illustrate some properties of the present technique, such as stability and the ability to propagate discontinuities.

Immersed boundary method for high-order flux reconstruction based on volume penalization

In the last decade, there has been a lot of interest in developing high-order methods as a viable option for unsteady scale-resolving-simulations which are increasingly important in the industrial design process. High-order methods offer the advantages of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. Despite significant advance in high-order solution methods, the general CFD workflow (geometry, CAD preparation, meshing, solution, post-processing) has largely remained unchanged, with mesh generation being a significant bottleneck and often determining the overall quality of the solution. In this work, we aim to combine the numerical advantages of the high-order Flux Reconstruction (FR) method and the simplicity of the mesh generation (or lack thereof) of the Immersed Boundary Method (IBM) for steady and unsteady problems over moving geometries. The volume penalization (penalty-IBM) method is selected for its ease of implementation and robustness. Detailed discussions about numerical implementation, including the boundary representation, mask function, data reconstruction, and selection of the penalization parameter are given. Advantages of combining volume penalization in the high-order framework are shown by various numerical test cases. The approach is firstly demonstrated for the linear advection-diffusion equation by investigating the numerical convergence for the coupled FR-IBM approach. Thereafter, the accuracy of the approach is demonstrated for canonical (static) test cases in 2D and 3D when compared to a standard body-fitted unstructured simulation. Finally, the efficiency of the method to handle moving geometries is demonstrated for the flow around an airfoil with pitching and plunging motions.

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

In this article we aim to approximate linear time fractional advection diffusion equations (TFADE) with Atangana-Baleanu- Caputo(ABC) derivative using local meshless method and Laplace transformation(LT). The method comprises of three steps. In the first step the the time variable is eliminated using LT. In the second step the reduced problem is solved using local meshless method. In the third step the solution of TFADE with ABC derivative is retrieved from local meshless methods solution by representing it as Bromwich integral. We then approximate the integral using some suitable quadrature rule. The stability and convergence of the method are discussed. The local meshless method is utilized to overcome the ill-conditioning issue of the interpolation matrices in global meshless methods and to over come the shape parameters sensitivity. Also in comparison with time stepping methods the LT is employed and contour integration technique is utilized to deal with the ABC derivative, which circumvent the calculation of costly convolution integrals in the approximation of ABC derivative, and also avoids the effect of time step on the stability and accuracy. Some test problems are considered in one and two dimensions to validate the proposed numerical method. The two dimensional problem is solved in regular and irregular domains. The computational experiments confirms that this method is computationally efficient and highly accurate for such type of problems.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

Peaceman Rachford Alternating Direct Implicitly Method for Linear Advection-Diffusion Equation and Its Application

In this paper, Peaceman-Rachford alternating direct implicitly methods presented and applied for solve linear advection-diffusion equation. First, the domain was discretized using the uniform mesh of step length and time step. Secondly, by applying the Taylor series methods, we discretize partial derivative of governing equation and we obtain the central difference equation for Partial differential equation of given governing equation in both duration. Then rearranging the obtained central difference equation; we write the two half scheme of the present method. From each half of these schemes, we obtain tri-diagonal coefficient matrices associated with the system of difference equation. Lastly by applying the Thomas algorithm and writing MATLAB code for the scheme we obtain solution of the governing linear advection diffusion equation. To validate the applicability of the proposed method, three model examples are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (L₁-norm) and L₂-norm, numerical error and experimental order of convergence. The stability and convergence of the present numerical method are also guaranteed and the comparability of numerical solution and the stability of the present method are presented by using the graphical and tabular form. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form with the standard and the split-form variants of the DGSEM. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

Numerical study for time fractional stochastic semi linear advection diffusion equations

In this work, a stochastic fractional advection diffusion model with multiplicative noise is studied numerically. The Galerkin finite element method in space and finite difference in time are used, where the fractional derivative is in Caputo sense. The error analysis is investigated via Galerkin finite element method. In terms of the Mittag Leffler function, the mild solution is obtained. For the error estimates, the strong convergence for the semi and fully discrete schemes are proved in a semigroup structure. Finally, two numerical examples are given to confirm the theoretical results.

Steady-state two-relaxation-time lattice Boltzmann formulation for transport and flow, closed with the compact multi-reflection boundary and interface-conjugate schemes

We introduce the steady-state two-relaxation-time (TRT) Lattice Boltzmann method. Owing to the symmetry argument, the bulk system and the closure equations are all expressed in terms of the equilibrium and non-equilibrium unknowns with the half discrete velocity set. The local mass-conservation solvability condition is adjusted to match the stationary, but also the quasi-stationary, solutions of the standard TRT solver. Additionally, the developed compact, boundary and interface-conjugate, multi-reflection (MR) concept preserves the efficient directional bulk structure and shares its parametrization properties. The method is exemplified in grid-inclined stratified slabs for two-phase Stokes flow and the linear advection-diffusion equation featuring the discontinuous coefficients and sources. The piece-wise parabolic benchmark solutions are matched exactly with the novel Dirichlet, pressure-stress, Neumann flux and Robin MR schemes. The popular, anti-bounce-back and shape-fitted Dirichlet continuity schemes are improved in the presence of both interface-parallel and perpendicular advection velocity fields. The steady-state method brings numerous advantages: it skips transient numerical instability, overpasses severe von Neumann parameter range limitations, tolerates high physical contrasts and arbitrary MR coefficients. The method is promising for faster computation of Stokes/Brinkman/Darcy linear flows in heterogeneous soil, but also heat and mass transfer problems governed by an accurate boundary and interface treatment.

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection-diffusion equations.

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection-diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.