Advection Method Research Articles

Exponentially fitted IMEX methods for advection–diffusion problems

The paper is devoted to the numerical solution of advection–diffusion problems of Boussinesq type, by means of adapted numerical methods. The adaptation occurs at two levels: along space, by suitably semidiscretizing the spatial derivatives through finite differences based on exponential fitting; along time integration, through an adapted IMEX method based on exponential fitting itself. Stability analysis is provided and numerical examples showing the effectiveness of the approach, also in comparison with the classical one, are given.

Explicit and implicit forms of differential quadrature method for advection–diffusion equation with variable coefficients in semi-infinite domain

In this paper, a numerical solution of one-dimensional time-dependent advection-diffusion equation with variable coefficients in semi-infinite domain is presented by using the differential quadrature method. Both the explicit and implicit approaches are provided. Totally, two solute dispersion problems are employed to simulate various conditions. The inhomogeneity of the domain is supplied by the spatially dependent flow. The problem domains are modeled with Chebyshev–Gauss–Lobatto grid points. In order to examine the accuracy and the efficiency of the suggested explicit and implicit approaches, analytical solutions, which are presented in the literature, are employed. In addition, the results of the above-mentioned method are compared with outcomes of the finite difference method. The results show that both of the explicit and implicit forms of the differential quadrature method are efficient, robust and reliable. But between these two forms, numerical predictions of implicit form are more accurate than explicit form.

Integrating Real-Time Fluid Simulation with a Voxel Engine

We present a method of adding sophisticated physical simulations to voxel-based games such as the hugely popular Minecraft (2012. http://minecraft.gamepedia.com/Liquid ), thus providing a dynamic and realistic fluid simulation in a voxel environment. An assessment of existing simulators and voxel engines is investigated, and an efficient real-time method to integrate optimized fluid simulations with voxel-based rasterisation on graphics hardware is demonstrated. We compare graphics processing unit (GPU) computer processing for a well-known incompressible fluid advection method with recent results on geometry shader-based voxel rendering. The rendering of visibility-culled voxels from fluid simulation results stored intermediately in CPU memory is compared with a novel, entirely GPU-resident algorithm.

Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity

We propose two tailored finite point methods for the advection---diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uniform convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.

Error estimates for transport problems with high Péclet number using a continuous dependence assumption

In this paper we discuss the behavior of stabilized finite element methods for the transient advection–diffusion problem with dominant advection and rough data. We show that provided a certain continuous dependence result holds for the quantity of interest, independent of the Péclet number, this quantity may be computed using a stabilized finite element method in all flow regimes. As an example of a stable quantity we consider the parameterized weak norm introduced in Burman (2014). The same results may not be obtained using a standard Galerkin method. We consider the following stabilized methods: Continuous Interior Penalty (CIP) and Streamline Upwind Petrov–Galerkin (SUPG). The theoretical results are illustrated by computations on a scalar transport equation with no diffusion term, rough data and strongly varying velocity field.

Gravity-wave effects on tracer gases and stratospheric aerosol concentrations during the 2013 ChArMEx campaign

Abstract. Coupled balloon-borne observations of Light Optical Aerosol Counter (LOAC), M10 meteorological global positioning system (GPS) sondes, ozonesondes, and GPS radio occultation data, are examined to identify gravity-wave (GW)-induced fluctuations on tracer gases and on the vertical distribution of stratospheric aerosol concentrations during the 2013 ChArMEx (Chemistry-Aerosol Mediterranean Experiment) campaign. Observations reveal signatures of GWs with short vertical wavelengths less than 4 km in dynamical parameters and tracer constituents, which are also correlated with the presence of thin layers of strong local enhancements of aerosol concentrations in the upper troposphere and the lower stratosphere. In particular, this is evident from a case study above Ile du Levant (43.02° N, 6.46° E) on 26–29 July 2013. Observations show a strong activity of dominant mesoscale inertia GWs with horizontal and vertical wavelengths of 370–510 km and 2–3 km respectively, and periods of 10–13 h propagating southward at altitudes of 13–20 km during 27–28 July. The European Centre for Medium-Range Weather Forecasts (ECMWF) analyses also show evidence of mesoscale inertia GWs with similar horizontal characteristics above the eastern part of France. Ray-tracing experiments indicate the jet-front system as the main source of observed GWs. Using a simplified linear GW theory, synthetic vertical profiles of dynamical parameters with large stratospheric vertical wind maximum oscillations of ±40 mms−1 are produced for the dominant mesoscale GW observed at heights of 13–20 km. Parcel advection method reveals signatures of GWs in the ozone mixing ratio and the tropospheric-specific humidity. Simulated vertical wind perturbations of the dominant GWs and small-scale perturbations of aerosol concentration (aerosol size of 0.2–0.7 µm) are revealed to be in phase in the lower stratosphere. Present results support the importance of vertical wind perturbations in the GW–aerosol relationship. Observed mesoscale GWs induce a strong modulation of the amplitude of tracer gases and the stratospheric aerosol background.

A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

This paper presents an in-depth numerical analysis of spatial fractional advection–diffusion equations (FADE) utilizing the finite element method (FEM). A traditional Galerkin finite element formulation of the pure fractional diffusion equation without advection may yield numerical oscillations in the solution depending on the fractional derivative order. These oscillations are similar to those that may arise in the integer-order advection–diffusion equation when using the Galerkin FEM. In a Galerkin formulation of a FADE, these oscillations are further compounded by the presence of the advection term, which we show can be characterized by a fractional element Peclet number that takes into account the fractional order of the diffusion term. To address this oscillatory behavior, a Petrov–Galerkin method is formulated using a fractional stabilization parameter to eliminate the oscillatory behavior arising from both the fractional diffusion and advection terms. A compact formula for an optimal fractional stabilization parameter is developed through a minimization of the residual of the nodal solution. Steady state and transient one-dimensional cases of the pure fractional diffusion and fractional advection–diffusion equations are implemented to demonstrate the effectiveness and accuracy of the proposed formulation.

A hybrid incremental projection method for thermal-hydraulics applications

A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya–Babuška–Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie–Chow interpolation or by using a Petrov–Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with 2 ≤ C F L ≤ 100 . The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e.g., in open-source packages such as HYPRE, PETSc, and Trilinos. • A new second-order hybrid finite-element/finite-volume projection algorithm for transient viscous flow has been introduced. • The hybrid discretization prevents pressure modes without using Rhie–Chow interpolation or a Petrov–Galerkin formulation. • A monotonicity-preserving advection method shown to deliver second-order accuracy on mixed element topology meshes. • Verification studies demonstrate hybrid projection solver accuracy and temporal stability for super-CFL conditions.

A Lagrangian-Eulerian advection scheme with moment-of-fluid interface reconstruction

ABSTRACTA Lagrangian–Eulerian advection scheme (LEAS) with Moment-of-Fluid (MoF) interface reconstruction is presented for interfacial flows. In MoF method, in addition to the volume fraction field, material centroid is used for interface reconstruction. Physically material centroid indicates the material location inside a mixed cell. Therefore, MoF method reconstructs linear interfaces exactly and is second order accurate for curved interfaces. Despite being second order accurate for static interface reconstruction, MoF reconstruction is inaccurate when centroid is not properly advected. An accurate centroid advection method based on Barycenter of centroid is discussed here. A comparison of this centroid advection with the pure Lagrangian advection based on Runge–Kutta integrator is demonstrated for two different advection tests. The superiority of MoF is established by comparing the volume error norm with other interface reconstructions. Significant improvement in accuracy is observed when LEAS material advection is used in conjuction with accurate centroid advection for MoF reconstruction.

WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

Abstract In this paper, we propose high-order numerical methods for the Riesz space fractional advection–dispersion equations (RSFADE) on a finite domain. The RSFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order α ∈ ( 0 , 1 ) and β ∈ ( 1 , 2 ] , respectively. Firstly, we utilize the weighted and shifted Grunwald difference operators to approximate the Riesz fractional derivative and present the finite difference method for the RSFADE. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of O ( τ 2 + h 2 ) . Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be O ( τ 4 + h 4 ) . Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent with theoretical analysis.

Krylov single-step implicit integration factor WENO methods for advection–diffusion–reaction equations

Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection–diffusion–reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection–diffusion–reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.

Quartic and quintic B-spline methods for advection–diffusion equation

Differential quadrature methods based on B-spline functions of degree four and five have been introduced to solve advection–diffusion equation numerically. Two initial-boundary value problems modeling the transportation of a concentration and distribution of an initial pulse are simulated using both methods. The errors of the numerical results obtained by both methods have been computed. Stability analysis for both methods is also studied by the use of matrix stability.

Compressive advection and multi‐component methods for interface‐capturing

SummaryThis paper develops methods for interface‐capturing in multiphase flows. The main novelties of these methods are as follows: (a) multi‐component modelling that embeds interface structures into the continuity equation; (b) a new family of triangle/tetrahedron finite elements, in particular, the P1DG‐P2(linear discontinuous between elements velocity and quadratic continuous pressure); (c) an interface‐capturing scheme based on compressive control volume advection methods and high‐order finite element interpolation methods; (d) a time stepping method that allows use of relatively large time step sizes; and (e) application of anisotropic mesh adaptivity to focus the numerical resolution around the interfaces and other areas of important dynamics. This modelling approach is applied to a series of pure advection problems with interfaces as well as to the simulation of the standard computational fluid dynamics benchmark test cases of a collapsing water column under gravitational forces (in two and three dimensions) and sloshing water in a tank. Two more test cases are undertaken in order to demonstrate the many‐material and compressibility modelling capabilities of the approach. Numerical simulations are performed on coarse unstructured meshes to demonstrate the potential of the methods described here to capture complex dynamics in multiphase flows. Copyright © 2015 John Wiley & Sons, Ltd.

Multilevel and local time-stepping discontinuous Galerkin methods for magma dynamics

Discontinuous Galerkin (DG) method is presented for numerical modeling of melt migration in a chemically reactive and viscously deforming upwelling mantle column at local chemical equilibrium. DG methods for both advection and elliptic equations provide a robust and efficient solution to the problems of melt migration in the asthenospheric upper mantle. Assembling and solving the elliptic equation is the major bottleneck in these computations. To address this issue, adaptive mesh refinement and local time-stepping methods have been proposed to improve the computational wall time. The robustness of DG methods is demonstrated through two benchmark problems by modeling detailed structure of high-porosity dissolution channels and compaction dissolution waves.

Batch advection for the piecewise linear vector field on simplicial grids

In practice, a sampled vector field is viewed as a piecewise linear field, which simplifies the interpolation of curve integration, and this assumption is widely adopted in most visualization algorithms. However, existing works omit the local coherent nature of this simplification, which could be leveraged to improve the efficiency of computation. In this paper, we reformulate the scheme of linear interpolation and integration for the simplicial mesh grid with homogeneous coordinates, and find that we can benefit significantly by simultaneously integrating multiple curves passing through a single cell. Based on this observation, we revise the 4th-order Runge–Kutta method and propose an efficient cell-wise and step-wise batch advection scheme for both 2D and 3D datasets, which is useful in the multiple particle tracing for the integral curves, e.g. streamline, pathline, timeline and streakline. We apply our method to some applications for a series of datasets, and the results show that our batch advection method can significantly speed up the curve integration while comparing with the traditional 4th-order Runge–Kutta.

Evaluation of the FTn Finite Volume Method for Transient Radiative Transfer in Anisotropically Scattering Medium

In this paper, we formulated, applied, and tested the FTn Finite Volume Method (FTn FVM) for transient radiative transfer in three-dimensional absorbing, emitting, and anisotropically scattering medium. Both the STEP and the Curved-Line Advection Method (CLAM) are introduced for spatial discretization of the transient radiative transfer equation. The results show that FTn FVM reduces largely the ray effects and it is more accurate than the standard FVM. Also, using both STEP and CLAM schemes, FTn FVM has smaller convergence time than the standard FVM for all cases. On the contrary, the STEP scheme is faster than the CLAM scheme but it has less accuracy. Then, the effects of optical thickness, scattering albedo, and anisotropy factor on incident radiation and radiative flux are presented and discussed.

Off the beaten path: a new approach to realistically model the orbital decay of supermassive black holes in galaxy formation simulations

We introduce a force correction term to better model the dynamical friction (DF) experienced by a supermassive black hole (SMBH) as it orbits within its host galaxy. This new approach accurately follows the orbital decay of a SMBH and drastically improves over commonly used advection methods. The force correction introduced here naturally scales with the force resolution of the simulation and converges as resolution is increased. In controlled experiments we show how the orbital decay of the SMBH closely follows analytical predictions when particle masses are significantly smaller than that of the SMBH. In a cosmological simulation of the assembly of a small galaxy, we show how our method allows for realistic black hole orbits. This approach overcomes the limitations of the advection scheme, where black holes are rapidly and artificially pushed toward the halo center and then forced to merge, regardless of their orbits. We find that SMBHs from merging dwarf galaxies can spend significant time away from the center of the remnant galaxy. Improving the modeling of SMBH orbital decay will help in making robust predictions of the growth, detectability, and merger rates of SMBHs, especially at low galaxy masses or at high redshift.

Correcting temporal sampling error in radar-rainfall: Effect of advection parameters and rain storm characteristics on the correction accuracy

Summary This study offers a method to correct for the radar temporal sampling error when determining radar-rainfall accumulations. The authors evaluate the correction effect with respect to multiple factors associated with storm advection, rainfall characteristics, and different rainfall accumulation time scales. The advection method presented in this study uses linear interpolation of static rain storm locations observed at two intermittent radar sampling times to correct for the missed rainfall accumulations. The advection correction is applied to the high space (0.5 km) and time (5-min) resolution radar-rainfall products provided by the Iowa Flood Center. We use the ground reference data from a high quality and high density rain gauge network distributed over the Turkey River basin in Iowa to evaluate the advection corrected rain fields. We base our evaluation on six rain events and examine the correction performance/improvement with respect to the advection discretization, spatial grid aggregation, rainfall basin coverage, and conditional average rainfall intensity. The results show that the 1-min advection discretization is sufficient to represent the observed distribution of storm velocities for the presented cases. Grid aggregation that is motivated by the need to expedite the computation may induce errors in estimating advection vectors. The authors found that while the advection correction tends to enhance the QPE accuracy for intense rain storms over small or isolated areas, it has little impact on the improvement of light rain estimation.

Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

We extend the positivity-preserving method of Zhang and Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge–Kutta (SSP-RK) time integration. Special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f; i.e., f∈[0,1]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments — including one example in spherical symmetry adopting the Schwarzschild metric — demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.

A new parallel subspace correction method for advection–diffusion equation

A new parallel subspace correction algorithm is proposed to solve advection–diffusion equation with mass-conservative characteristic finite element (MCC-FE) procedure. The dependence relations of the subdomains overlapping size, spacial mesh parameter, time step, iteration number with the convergence rate is analyzed, and the a priori error estimate of this parallel algorithm is given. Some numerical experiments are given to verify our theoretical result.