High-order schemes for 2D unsteady biogeochemical ocean models

Accurate numerical modeling of biogeochemical ocean dynamics is essential for numerous applications, including coastal ecosystem science, environmental management and energy, and climate dynamics. Evaluating computational requirements for such often highly nonlinear and multiscale dynamics is critical. To do so, we complete comprehensive numerical analyses, comparing low- to high-order discretization schemes, both in time and space, employing standard and hybrid discontinuous Galerkin finite element methods, on both straight and new curved elements. Our analyses and syntheses focus on nutrient–phytoplankton–zooplankton dynamics under advection and diffusion within an ocean strait or sill, in an idealized 2D geometry. For the dynamics, we investigate three biological regimes, one with single stable points at all depths and two with stable limit cycles. We also examine interactions that are dominated by the biology, by the advection, or that are balanced. For these regimes and interactions, we study the sensitivity to multiple numerical parameters including quadrature-free and quadrature-based discretizations of the source terms, order of the spatial discretizations of advection and diffusion operators, order of the temporal discretization in explicit schemes, and resolution of the spatial mesh, with and without curved elements. A first finding is that both quadrature-based and quadrature-free discretizations give accurate results in well-resolved regions, but the quadrature-based scheme has smaller errors in under-resolved regions. We show that low-order temporal discretizations allow rapidly growing numerical errors in biological fields. We find that if a spatial discretization (mesh resolution and polynomial degree) does not resolve the solution, oscillations due to discontinuities in tracer fields can be locally significant for both low- and high-order discretizations. When the solution is sufficiently resolved, higher-order schemes on coarser grids perform better (higher accuracy, less dissipative) for the same cost than lower-order scheme on finer grids. This result applies to both passive and reactive tracers and is confirmed by quantitative analyses of truncation errors and smoothness of solution fields. To reduce oscillations in un-resolved regions, we develop a numerical filter that is active only when and where the solution is not smooth locally. Finally, we consider idealized simulations of biological patchiness. Results reveal that higher-order numerical schemes can maintain patches for long-term integrations while lower-order schemes are much too dissipative and cannot, even at very high resolutions. Implications for the use of simulations to better understand biological blooms, patchiness, and other nonlinear reactive dynamics in coastal regions with complex bathymetric features are considerable.     

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other authors.     

Numerical modelling of coupled surface and subsurface flow systems

This paper presents and compares several numerical solutions of the coupled system of Navier–Stokes and Darcy equations. The schemes are based on combinations of the finite element method and the discontinuous Galerkin method. Accuracy and robustness of the methods are investigated for heterogeneous porous media. The importance of local mass conservation for filtration problems is also discussed.     

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes

We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic three-dimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward.     

A two-grid method for coupled free flow with porous media flow

This paper presents a two-grid method for solving systems of partial differential equations modelling incompressible free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.     

Application of the discontinuous spectral Galerkin method to groundwater flow

The discontinuous spectral Galerkin method uses a finite-element discretization of the groundwater flow domain with basis functions of arbitrary order in each element. The independent choice of the basis functions in each element permits discontinuities in transmissivity in the flow domain. This formulation is shown to be of high order accuracy and particularly suitable for accurately calculating the flow field in porous media. Simulations are presented in terms of streamlines in a bidimensional aquifer, and compared with the solution calculated with a standard finite-element method and a mixed finite-element method. Numerical simulations show that the discontinuous spectral Galerkin approximation is more efficient than the standard finite-element method (in computing fluxes and streamlines/pathlines) for a given accuracy, and it is more accurate on a given grid. On the other hand the mixed finite-element method ensures the continuity of the fluxes at the cell boundaries and it is particular efficient in representing complicated flow fields with few mesh points. Simulations show that the mixed finite-element method is superior to the discontinuous spectral Galerkin method producing accurate streamlines even if few computational nodes are used. The application of the discontinuous Galerkin method is thus of interest in groundwater problems only when high order and extremely accurate solutions are needed.     

地震波动方程的局部间断有限元方法数值模拟

近年来,随着地震波数值模拟对计算精度和效率的要求越来越高,间断有限元方法开始受到越来越多的关注.本文中,针对具有吸收边界条件的二维地震声波波动方程,作者提出了一种基于局部间断有限元方法的数值模拟算法.该算法在空间上使用局部间断有限元方法进行离散,在时间上采用了显式蛙跳格式.在这种时空离散的组合方式下,每个时间步上,此算法在空间剖分的每个单元上的求解计算是相互独立的,因而具有极高的并行性.通过数值算例,我们将该算法与连续有限元方法进行了比较.结果表明,本算法不仅具有对起伏构造的良好适应性,而且在计算效率和计算精度等方面,都具有优越性.     

Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes.     

Mass lumping models of the linear diffusion equation

The nodal domain integration method is used to develop a numerical model of the linear diffusion equation. The nodal domain integration approach is shown to represent an infinity of finite element mass matrix lumping schemes including the Galerkin and subdomain integration versions of the weighted residual method and an integrated finite difference method. Neumann, Dirichlet and mixed boundary conditions are accommodated analogous to the Galerkin finite element method. In order to reduce the overall integrated approximation relative error, a mass matrix lumping formulation is developed which is based on the Crank-Nicolson time advancement approximation. The optimum mass lumping factors are found to be strongly related to the model timestep size.     

Application of modified Patankar schemes to stiff biogeochemical models for the water column

In this paper, we apply recently developed positivity preserving and conservative Modified Patankar-type solvers for ordinary differential equations to a simple stiff biogeochemical model for the water column. The performance of this scheme is compared to schemes which are not unconditionally positivity preserving (the first-order Euler and the second- and fourth-order Runge–Kutta schemes) and to schemes which are not conservative (the first- and second-order Patankar schemes). The biogeochemical model chosen as a test ground is a standard nutrient–phytoplankton–zooplankton–detritus (NPZD) model, which has been made stiff by substantially decreasing the half saturation concentration for nutrients. For evaluating the stiffness of the biogeochemical model, so-called numerical time scales are defined which are obtained empirically by applying high-resolution numerical schemes. For all ODE solvers under investigation, the temporal error is analysed for a simple exponential decay law. The performance of all schemes is compared to a high-resolution high-order reference solution. As a result, the second-order modified Patankar–Runge–Kutta scheme gives a good agreement with the reference solution even for time steps 10 times longer than the shortest numerical time scale of the problem. Other schemes do either compute negative values for non-negative state variables (fully explicit schemes), violate conservation (the Patankar schemes) or show low accuracy (all first-order schemes).     

Coupling local discontinuous and continuous galerkin methods for flow problems

In this paper, we discuss the local discontinuous Galerkin (LDG) method applied to elliptic flow problems and give details on its implementation, focusing specifically on the case of piecewise linear approximating functions. The LDG method is one a family of discontinuous Galerkin (DG) methods proposed for diffusion models. These DG methods allow for very general hp finite element meshes, and produce locally conservative fluxes which can be used in coupling flow with transport. The drawback to DG methods, when compared to their continuous counterparts, is the number of degrees of freedom required to compute the solution. This motivates a coupled approach, discussed herein, where the solution is allowed to be continuous or discontinuous on a node-by-node basis. This coupled approximation is locally conservative in regions where the numerical solution is discontinuous. Numerical results for fully discontinuous, continuous and coupled discontinuous/continuous solutions are given, where we compare solution accuracy, matrix condition numbers and mass balance errors for the various approaches.     

High-order <Emphasis Type="Italic">h</Emphasis>-adaptive discontinuous Galerkin methods for ocean modelling

In this paper, we present an h-adaptive discontinuous Galerkin formulation of the shallow water equations. For a discontinuous Galerkin scheme using polynomials up to order , the spatial error of discretization of the method can be shown to be of the order of , where is the mesh spacing. It can be shown by rigorous error analysis that the discontinuous Galerkin method discretization error can be related to the amplitude of the inter-element jumps. Therefore, we use the information contained in jumps to build error metrics and size field. Results are presented for ocean modelling problems. A first experiment shows that the theoretical convergence rate is reached with the discontinuous Galerkin high-order h-adaptive method applied to the Stommel wind-driven gyre. A second experiment shows the propagation of an anticyclonic eddy in the Gulf of Mexico. An erratum to this article can be found at     

A finite element sea ice model of the Canadian Arctic Archipelago

The Canadian Arctic Archipelago (CAA) is a complex area formed by narrow straits and islands in the Arctic. It is an important pathway for freshwater and sea-ice transport from the Arctic Ocean to the Labrador Sea and ultimately to the Atlantic Ocean. The narrow straits are often crudely represented in coupled sea-ice–ocean models, leading to a misrepresentation of transports through these straits. Unstructured meshes are an alternative in modelling this complex region, since they are able to capture the complex geometry of the CAA. This provides higher resolution in the flow field and allows for more accurate transports (but not necessarily better modelling). In this paper, a finite element sea-ice model of the Arctic region is described and used to estimate the sea-ice fluxes through the CAA. The model is a dynamic–thermodynamic sea-ice model with elastic–viscous–plastic rheology and is coupled to a slab ocean, where the temperature and salinity are restored to climatology, with no velocities and surface elevation. The model is spun-up from 1973 to 1978 with NCEP/NARR reanalysis data. From 1979 to 2007, the model is forced by NCEP/DoE reanalysis data. The large scale sea-ice characteristics show good agreement with observations. The total sea-ice area agrees very well with observations and shows a sensitivity to the Arctic oscillation (AO). For 1998–2002, we find estimates for the sea-ice volume and area fluxes through Admunsen Gulf, McClure Strait and the Queen Elizabeth Islands that compare well with observation and are slightly better than estimates from other models. For Nares Strait, we find that the fluxes are much lower than observed, due to the missing effect of topographic steering on the atmospheric forcing fields. The 1979–2007 fluxes show large seasonal and interannual variability driven primarily by variability in the ice velocity field and a sensitivity to the AO and other large-scale atmospheric variability, which suggests that accurate atmospheric forcing might be crucial to modelling the CAA.     

Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study

This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater.     

Open boundary value problems in ocean dynamics by finite elements

This paper presents recent results of application of the finite element models to wave overtopping and wave run-up problems in ocean dynamics. Open boundaries are prescribed as natural boundary condition obtained from the continuity equation of the Galerkin finite element formulation. The numerical results are, in general, reasonably good agreements with the histrical field data.     

Simulation of long-term transient density-dependent transport in groundwater

A numerical model for the economical simulation of long-term transient response in density-dependent transport problems is introduced. Although a classical Galerkin finite element approach is used, emphasis on optimum efficiency throughout the development results in a scheme that is found to be significantly less costly than comparable existing schemes. This advantage in efficiency increases the scope of simulation problems that can be handled within the constraints of a limited research budget. Some distinctive aspects are the elimination of static quantities in the fluid continuity equation, achieved by the introduction of equivalent freshwater head, and the elimination of numerical integration, achieved by the deliberate choice of linear elements. As a result of this choice, fluid velocities are discontinuous across the element boundaries. It is shown, however, that the solution obtained with discontinuous velocities approaches that obtained with continuous velocities as the grid is refined, and that the two types of solutions give essentially the same results when the elements are in the same size range. The model is applied to simulate the complete transient response for a well-known problem of seawater intrusion in a confined aquifer. The simulation is performed with both the constant dispersion coefficient used by previous researchers, and a more physically realistic velocity-dependent dispersion coefficient. Responses are found to be substantially different for the two types of coefficients, with the velocity-dependent dispersion coefficient producing much slower convergence to a state of dynamic equilibrium, and a much more pointed saltwater toe, which at the bottom of the aquifer tends to a sharp interface at equilibrium. Finally, it is shown by means of large-scale applications that the model is capable of efficiently simulating the long-term transient response in systems of practical significance.     

An Hermitian finite element solution of the two-dimensional saturated-unsaturated flow equation

This paper describes a Galerkin-type finite element solution of the two-dimensional saturated-unsaturated flow equation. The numerical solution uses an incomplete (reduced) set of Hermitian cubic basis functions and is formulated in terms of normal and tangential coordinates. The formulation leads to continuous pressure gradients across interelement boundaries for a number of well-defined element configurations, such as for rectangular and circular elements. Other elements generally lead to discontinuous gradients; however, the gradients remain uniquely defined at the nodes. The method avoids calculation of second-order derivatives, yet retains many of the advantages associated with Hermitian elements. A nine-point Lobatto-type integration scheme is used to evaluate all local element integrals. This alternative scheme produces about the same accuracy as the usual 9- or 16-point Gaussian quadrature schemes, but is computationally more efficient.     

Direct Conservative Domain in the Continuous Galerkin Method for Groundwater Models

The continuous Galerkin finite element method is commonly considered locally nonconservative because a single element with fluxes computed directly from its potential distribution is unable to conserve its mass and fluxes across edges that are discontinuous. Some literature sources have demonstrated that the continuous Galerkin method can be locally conservative with postprocessed fluxes. This paper proposes the concept of a direct conservative domain (DCD), which could conserve mass when fluxes are computed directly from the potential distribution. Also presented here is a method for modifying the advection fluxes to obtain different conservative domains from the DCDs. Furthermore, DCDs are used to analyze the local conservation of several postprocessing algorithms, for which DCDs provide the theoretical basis. The local conservation of DCDs and the proposed method are illustrated and verified by using a hypothetical 2‐D model.     

An efficient numerical model for multicomponent compressible flow in fractured porous media

An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency.     

A generalized least-squares family of algorithms for transient dynamic analysis

By use of the generalized least-squares procedure, in conjunction with a finite element approximation in time, a simple three-time-level family of time integration schemes is derived. This results in fourth-order accurate unconditionally stable algorithms and stable eighth-order accurate non-dissipative algorithms. Numerical examples show the accuracy of the proposed schemes in comparison with the Fox-Goodwin formula and Newmark's average acceleration method.