Advection Method Research Articles

Three-dimensional FTn finite volume solution of short-pulse laser propagation through heterogeneous medium

In this paper, 3-D heterogeneous medium, containing small inhomogeneous zones, subjected to a short-pulse laser has been examined by solving the transient radiative transfer equation. Both curved-line advection method and STEP schemes of the FTn finite volume method have been applied. The curved-line advection method predictions proved that a decrease of the false scattering and ray effects are obtained. In fact, there was a good agreement between the curved-line advection method and the Monte Carlo method. However, the STEP results are slightly mismatching the predictions of the aforementioned reference method. Then, the effects of the absorption coefficient, the size, the number and the position of inhomogeneous zone on the transmittance and reflectance signals have been analyzed. The predictions showed that the increase of the size of the inhomogeneity reduces the intensity of radiation. For both homogenous and heterogonous medium, the change of the detector position varies both the broadening of the signal pulse-width and the time with peak reflectance and/or transmittance. That is can be explained by the effects of the distance and the medium property between the laser-incident source and the detector position. Thus, these both parameters are the main factors for determining the peak position and the pulse broadening. Finally, the effects of the absorption coefficient in the inhomogeneity zone on the absolute values of logarithmic slope has been discussed. The results proved that the absolute values of logarithmic slope may be a perfect indicator for detecting any abnormal absorbing zones in the medium.

An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

In this paper, a linearized L1 Legendre–Galerkin spectral method for the two-dimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of non-smooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method.

Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible Stokes flow

SUMMARY Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations. On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy. In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order. Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark in aspect (the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.

Forecasting short-term dynamics of shallow cumuli using dynamic mode decomposition

The application of dynamic mode decomposition to clear-sky index forecasting of shadowing effects of convective fair-weather cumulus clouds is presented. Cloud dynamics are captured by sequences of visible-light photographic video frames. This method can be more easily applied to the modeling of cloud evolution than traditional fluid-based methods and can enhance the existing frozen-cloud advection methods. Its use is demonstrated for an actual fair-weather cumulus cloud image sequence and compared to an advection-only forecast. It is concluded that the method shows promise for very short-term clear-sky index forecasting for up to 7 minute horizons.

A diagonal splitting method for solving semidiscretized parabolic partial differential equations

AbstractIn this work, a diagonal splitting idea is presented for solving linear systems of ordinary differential equations. The resulting methods are specially efficient for solving systems which have arisen from semidiscretization of parabolic partial differential equations (PDEs). Unconditional stability of methods for heat equation and advection–diffusion equation is shown in maximum norm. Generalization of the methods in higher dimensions is discussed. Some illustrative examples are presented to show efficiency of the new methods.

High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems

We construct in this paper two efficient spectral algorithms in the frequency space for solving unsteady advection–reaction–diffusion equations with constant and variable coefficients. We first consider a Jacobi–Galerkin method for solving linear equations with constant coefficients. We then develop the direct solution algorithm for the linear advection–reaction–diffusion equations with variable coefficients using the Jacobi–Galerkin method with numerical integration. The proposed Jacobi–Galerkin methods, both in temporal and spatial discretizations, are successfully developed to handle the two-dimensional unsteady advection–reaction–diffusion equations with constant and variable coefficients and with fractional orders. In these methods, the model solution is expanded in both space and time in terms of polynomials bases built upon a linear combination of Jacobi polynomials. The homogeneous initial and Dirichlet boundary conditions are satisfied exactly by expanding the model solution in terms of these polynomials. The proposed Jacobi–Galerkin methods yield an exponential rate of convergence when the solution is smooth and allow a great flexibility to handle multi-dimensional time fractional advection–reaction–diffusion equations. Finally, a series of numerical examples are presented to demonstrate the efficiency and flexibility of the methods.

Filter regularization method for a time-fractional inverse advection\u2013dispersion problem

A filter regularization method is developed to solve a time-fractional inverse advection–dispersion problem, which is based on the modified ‘kernel’ idea. Proofs of convergence are given under both priori and posteriori regularization parameter choice rules. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.

Simulation of glioblastoma growth using a 3D multispecies tumor model with mass effect.

In this article, we present a multispecies reaction-advection-diffusion partial differential equation coupled with linear elasticity for modeling tumor growth. The model aims to capture the phenomenological features of glioblastoma multiforme observed in magnetic resonance imaging (MRI) scans. These include enhancing and necrotic tumor structures, brain edema and the so-called "mass effect", a term-of-art that refers to the deformation of brain tissue due to the presence of the tumor. The multispecies model accounts for proliferating, invasive and necrotic tumor cells as well as a simple model for nutrition consumption and tumor-induced brain edema. The coupling of the model with linear elasticity equations with variable coefficients allows us to capture the mechanical deformations due to the tumor growth on surrounding tissues. We present the overall formulation along with a novel operator-splitting scheme with components that include linearly-implicit preconditioned elliptic solvers, and a semi-Lagrangian method for advection. We also present results showing simulated MRI images which highlight the capability of our method to capture the overall structure of glioblastomas in MRIs.

On algebraic TVD-VOF methods for tracking material interfaces

We revisit simple algebraic VOF methods for advection of material interfaces based of the well established TVD paradigm. We show that greatly improved representation of contact discontinuities is obtained through use of a novel CFL-dependent limiter whereby the classical TVD bounds are exceeded. Perfectly crisp numerical interfaces are obtained with very limited numerical atomization (flotsam and jetsam) as compared to previous SLIC schemes. Comparison of the algorithm with accurate geometrical VOF shows larger error at given mesh resolution, but comparable efficiency when the reduced computational cost is accounted for.

A numerical method for a time-fractional advection–dispersion equation with a nonlinear source term

In this paper we propose an implicit finite-difference scheme to approximate the solution of an initial-boundary value problem for a time-fractional advection–dispersion equation with variable coefficients and a nonlinear source term. The time fractional derivative is taken in the sense of Caputo. The method is unconditionally stable and convergent. Some numerical examples are included and the results confirm the theoretical analysis. One of the examples is the fractional Fisher equation of mathematical biology.

The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

Finite element modeling of nonlinear reaction–diffusion–advection systems of equations

PurposeThis paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed.Design/methodology/approachA stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation.FindingsThe stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model.Originality/valueA stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

A Crank–Nicolson ADI Galerkin–Legendre spectral method for the two-dimensional Riesz space distributed-order advection–diffusion equation

In the paper, a Crank–Nicolson alternating direction implicit (ADI) Galerkin–Legendre spectral scheme is presented for the two-dimensional Riesz space distributed-order advection–diffusion equation. The Gauss quadrature has a higher computational accuracy than the mid-point quadrature rule, which is proposed to approximate the distributed order Riesz space derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Galerkin–Legendre spectral scheme and in time using the Crank–Nicolson difference method. Stability and convergence analysis are verified for the numerical approximation. A lot of numerical results are demonstrated to justify the theoretical analysis.

Blast wave propagation in survival shelters: experimental analysis and numerical modelling

The propagation of blast and shock waves in confined environments is a complex phenomenon; yet, being able to derive valid predictions of such phenomena is highly relevant, for example, when it comes to the assessment of protection of personnel in military environments. This study looks at the propagation of blast waves inside a compound survival shelter. Experimental analyses are performed on a small-scale model of the actual configuration of the shelter subjected to the detonation of an explosive charge at different locations close to its entrance. Pressure-time signals are recorded on a number of locations in the model. A numerical model is also developed to complement the experimental programme, based on the explicit finite element (FE) code LS-DYNA. The recorded experimental data (e.g., pressure and impulse) are compared with the numerical predictions to validate the FE model. The authors discuss two different modelling approaches (the Lagrangian and the MM-ALE formulations) and analyse the influence of using a different number of ambient layers, the advection method, the time-step size, and level of discretisation. The proposed numerical model predicts and captures the relevant stages of the propagation of the shock wave very well, with error levels on the resulting specific impulse always lower than 19% when compared to the experimental observations.

The Fargo Method for Keplerian Advection: Error Analysis and Correction

Abstract The Fargo method alleviates the time-step constraint due to Keplerian advection by shifting flow quantities an integer number of grid intervals in the azimuthal direction. We point out that the implied change of the azimuthal coordinate means that all radial derivatives must carry an extra operator, , which is not present in current implementations and therefore implies an error. Here, t 0 is time at the beginning of a time step, is the rotation rate in the coordinate transformation, and ϕ is the azimuthal coordinate. If the quantity were smooth, the factor would imply that the error is first-order in time. However, integerization in the original Fargo scheme means that is a δ-function wherever suffers a jump, so the error becomes unbounded (but more localized) in a finite-difference or finite-volume setting as the radial grid size . Numerical tests showed that the original Fargo method produces noticeable errors where has a jump. These errors are advected by the flow into the rest of the domain. Even if integerization were not employed, the Fargo code is second-order in time and evaluates transport terms at therefore, the extra operator still cannot be neglected. The correction proposed here is to perform a continuous-in-r shift at the end of each time step using a fast Fourier transform and to include the additional operator. Simulation tests including cpu times are provided for a scheme that is fourth-order in space and time.

Spatially adaptive long-term semi-Lagrangian method for accurate velocity advection

We introduce a new advection scheme for fluid animation. Our main contribution is the use of long-term temporal changes in pressure to extend the commonly used semi-Lagrangian scheme further back along the time axis. Our algorithm starts by tracing sample points along a trajectory following the velocity field backwards in time for many steps. During this backtracing process, the pressure gradient along the path is integrated to correct the velocity of the current time step. We show that our method effectively suppresses numerical diffusion, retains small-scale vorticity, and provides better long-term kinetic energy preservation.

SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations

We present the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime.According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients.The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

Multiscale analysis of heterogeneous domain decomposition methods for time-dependent advection–reaction–diffusion problems

Domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods. We are interested here in the case where there is an accurate but expensive model one should use in the entire domain, but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected. For the model problem of a time dependent advection–reaction–diffusion equation in one spatial dimension, we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain. Using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain, we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non-variational heterogeneous domain decomposition methods. We show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis, and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity.

Reconstructed discontinuous Galerkin methods for linear advection–diffusion equations based on first-order hyperbolic system

Newly developed reconstructed Discontinuous Galerkin (rDG) methods are presented for solving linear advection–diffusion equations on hybrid unstructured grids based on a first-order hyperbolic system (FOHS) formulation. Benefiting from both FOHS and rDG methods, the developed hyperbolic rDG methods are reliable, accurate, efficient, and robust, achieving higher orders of accuracy than conventional DG methods for the same number of degrees-of-freedom. Superior accuracy is achieved by reconstruction of higher-order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least-squares/variational reconstruction. Unsteady capability is demonstrated by an L-stable implicit time-integration scheme. A number of advection–diffusion test cases with a wide range of Reynolds numbers, including boundary layer type problems and unsteady cases, are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous grids, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the linear advection–diffusion equations.

Finite Difference Method for Time-Space Fractional Advection-Diffusion Equations with Riesz Derivative.

In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.