Large Courant Numbers Research Articles

Treatment for the Singularity Problem in the Phase Segregation

In the Eulerian–Eulerian two-fluid model, comprising air and droplets, singularities often arise as the phase fraction of dispersed droplets approaches zero. The nonconservative momentum equations can address this issue, but it will lead to inaccurate solutions when encountering discontinuities. To tackle the singularity problem and ensure accurate solutions for the droplet field, a novel adaptive form of momentum equations is proposed. The adaptive form defaults to the conservative form and smoothly transitions to the phase-intensive form when the phase fraction falls below a critical value, specifically 1/1000 of the inlet. The LU-SGS algorithm, enhanced with the inner iteration step method, is employed for numerical solutions of the droplet field. Comparative analysis reveals that the discrepancy between the results obtained from the adaptive form and the conservative form is negligible, under 0.5‰. Notably, with larger Courant numbers, the conservative form diverges from waterless regions despite imposing the minimum phase fraction constraint. Conversely, the adaptive form consistently maintains a Courant number of five times or more, potentially exceeding a hundredfold in complex flows. Although the adaptive form moderately increases complexity and computational time for a single time step, the improvement of the maximum Courant number will bring significant computational efficiency advantages. Moreover, the adaptive form holds promise for application in various other multiphase flow scenarios.

An efficient forward semi-Lagrangian model

An efficient forward trajectory model is proposed, in which the property and position of the fluids advected from the Euler coordinates to the Lagrangian coordinates can be accurately evaluated. After sorting and aligning those fluid elements on the irregular Lagrangian curves, we apply the cubic or other high-degree polynomials to interpolate the properties of the elements from the irregular curves to the regular grids. There is no need to solve the cubic equations and the associated coefficients as proposed previously. The model is quite simple, accurate, and much more efficient than the previous models. It also allows higher-order polynomials to be employed in the interpolations. It is suitable for simulating the multi-dimensional fast-moving flows with large Courant Numbers, the transport of pollutants in the atmosphere and ocean, and movement of raindrops in atmospheric models.

A novel high accuracy finite-difference time-domain method

The finite-difference time-domain (FDTD) method is widely used for numerical simulations of electromagnetic waves and acoustic waves. It is known, however, that the Courant condition is restricted in higher dimensions and with higher order differences in space. Although it is possible to relax the Courant condition by utilizing the third-degree difference in space, there remains a large anisotropy in the numerical dispersion at large Courant numbers. This study aims to reduce the anisotropy in the numerical dispersion and relax the Courant condition simultaneously. A new third-degree difference operator including the Laplacian is introduced to the time-development equations of FDTD(2,4) with second- and fourth-order accuracies. The present numerical simulations have demonstrated that numerical oscillations due to the anisotropic dispersion relation are reduced with the new operator.Graphical

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

A compatible finite element discretisation for the nonhydrostatic vertical slice equations

We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.

Relaxation of the Courant Condition in the Explicit Finite-Difference Time-Domain Method With Higher-Degree Differential Terms

A new explicit and nondissipative finite-difference time-domain (FDTD) method in two and three dimensions is proposed for the relaxation of the Courant condition. The third-degree spatial difference terms with second- and fourth-order accuracies are added with coefficients to the time-development equations of FDTD(2,4). Optimal coefficients are obtained by a brute-force search of the dispersion relations, which reduces phase velocity errors but satisfies the numerical stabilities as well. The new method is stable with large Courant numbers, whereas the conventional FDTD methods are unstable. The new method also has smaller numerical errors in the phase velocity than conventional FDTD methods with small Courant numbers.

A three‐dimensional positivity‐preserving and conservative multimoment finite‐volume transport model on a cubed‐sphere grid

AbstractA three‐dimensional positivity‐preserving and conservative transport model on a cubed‐sphere grid is presented using a multimoment finite‐volume (MMFV) method. The three‐point fourth‐order MMFV method with boundary gradient switching projection is utilized to ensure nonoscillatory numerical solutions in the horizontal, while the simple and computationally efficient semi‐Lagrangian piecewise rational method is used to allow a large time step in the vertical. By means of horizontal and vertical separation, a Strang‐type splitting technique is adopted to advance the numerical solutions in time. Non‐negative corrections are made to achieve positive‐definite preservation exactly. The resulting transport model is inherently conservative, high‐accuracy, and positive‐definite, and allows a large Courant number no matter how dense the vertical grid is. Various benchmark test cases, including three representative advection tests and an idealized terminator chemistry test, are conducted to validate the performance of the model presented. The numerical results indicate that the model is quite competitive in comparison with the existing models in the literature and looks very promising in real atmospheric simulations.

A Stable and High Accurate Numerical Simulator for Convection-Diffusion Equation

This study presents a simulator to obtain numerical solution of convection-diffusion equation. It includes explicit, fully implicit and semi-implicit time discretization techniques. Although the explicit method is efficient for some simple conduction problems, it needs stability criteria. Otherwise, it is unstable especially for high Courant number and large time step size. When using the explicit method, the time step size is chosen with care to get well-posed numerical solution. This requirement sets a serious constraint for the explicit method because choosing small time step size causes the simulation time to become quite long. Even though the implicit method is stable for large time step size and high Courant number, it leads to numerical dispersion like the explicit method. On the other hand, second-order accurate semi-implicit method significantly reduces numerical dispersion. In addition to time discretization techniques, this simulator contains several space discretization methods such as first-order upstream and UMIST (University of Manchester Institute of Science and Technology) techniques. The proposed numerical simulator is suitable for easily using the different combinations of time and space discretization methods. Secondly, this study is to propose the use of semi-implicit time discretization technique with UMIST space discretization method to minimize numerical dispersion and suppress unphysical oscillation. In spite of the fact that the UMIST method suppresses unphysical oscillation, it causes a small and undesired oscillation at flood front for very large Courant number. Third objective of this study is to propose minor modification on the UMIST method to eliminate this unphysical oscillation.

Efficient implementation of characteristic-based schemes on unstructured triangular grids

Using characteristics to treat advection terms in time-dependent PDEs leads to a class of schemes, e.g., semi-Lagrangian and Lagrange–Galerkin schemes, which preserve stability under large Courant numbers, and may therefore be appealing in many practical situations. Unfortunately, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids, and thus prevent the use of large time-step schemes on complex geometries. In this paper, we perform an in-depth analysis of the main recipes available for characteristic location, and propose a technique to improve the efficiency of this phase, using additional information related to the advecting vector field. This results in a clear improvement of execution times in the unstructured case, thus extending the range of applicability of large time-step schemes.

The suite of Taylor–Galerkin class schemes for ice transport on sphere implemented by the INMOST package

AbstractRealizations of the numerical solution of the scalar transport equation on the sphere, written in divergent form, are presented. Various temporal discretizations are considered: the one-step Taylor–Galerkin method (TG2), the two-step Taylor–Galerkin method of the second (TTG2), third (TTG3), and fourth (TTG4) orders. The standard Finite-Element Galerkin method with linear basis functions on a triangle is applied as spatial discretization. The flux correction technique (FCT) is implemented. Test runs are carried out with different initial profiles: a function fromC∞(Gaussian profile) and a discontinuous function (slotted cylinder). The profiles are advected by reversible, nondivergent velocity fields, therefore the initial distribution coincides with the final one. The case of a divergent velocity field is also considered to test the conservation and positivity properties of the schemes. It is demonstrated that TG2, TTG3, and TTG4 schemes with FCT applied give the best result for small Courant numbers, and TTG2, TTG4 are preferable in case of large Courant number. However, TTG2+FCT scheme has the worst stability. The use of FCT increases the integral errors, but ensures that the solution is positive with high accuracy. The implemented schemes are included in the dynamic core of a new sea ice model developed using the INMOST package. The acceleration of the parallel program and solution convergence with spatial resolution are demonstrated.

A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations

We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr>1); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present a detailed stability and convergence analyses for the exponential Euler DG scheme.

A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid. Two kinds of moments (i.e., point values (PV moment) at cell interfaces and volume integrated average (VIA moment) value) are defined within a single cell. The PV moment is updated by a conventional semi-Lagrangian method, while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation. Different from the spatial approximation used in the CSL2 (conservative semi-Lagrangian scheme with second order polynomial function) scheme, a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment. To achieve exactly positive-definite preserving, two kinds of corrections are made on the original conservative semi-Lagrangian with rational function (CSLR) scheme. The resulting scheme is inherently conservative, non-negative, and allows a Courant number larger than one. Moreover, the spatial reconstruction can be performed within a single cell, which is very efficient and economical for practical implementation. In addition, a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry, which benefitsthe implementation of the 1D CSLR solver with large Courant number. The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry. Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical non-negativity, and it has the potential to transport the tracers accurately in a real atmospheric model.

Coastal pollutant transport modeling using smoothed particle hydrodynamics with diffusive flux

The Smoothed Particle Hydrodynamics (SPH) method is used in this paper to simulate the transport process of coastal pollutant by solving the two-dimensional (2D) depth averaged advection-diffusion equation in Lagrangian framework. To avoid directly discretizing the second-order derivatives, the diffusion terms are decomposed into two first-order derivatives by introducing the diffusive flux. To verify the proposed Lagrangian particle model, numerical experiments on four typical tests are performed, including the discontinuity, pure advection, grid anisotropy and large deforming flow. The high-resolution Total Variation Diminishing (TVD) scheme - the modified TVD Lax–Friedrichs method with Superbee limiter (MTVDLF-Superbee) - is also implemented for detailed comparison. The simulation results indicated that the present SPH-based model has better performance than MTVDLF-Superbee. In addition, the stability, convergence and efficiency of the proposed model are numerically analyzed. The Lagrangian particle model with diffusive flux gives satisfactory results with big time step and hence allows large Courant number, guaranteeing the computational efficiency.

SLIC: A Semi-Lagrangian Implicitly Corrected method for solving the compressible Euler equations

The Semi-Lagrangian (SL) scheme is a popular method used in weather and climate models due to its ability to handle the large Courant numbers encountered in polar regions and jet-streams. This makes SL-based models highly efficient since relatively large time steps can be employed especially when coupled with a semi-implicit scheme for treating the fast acoustic and gravity waves. For problems where the velocity field is unknown in advance and is part of the solution either an extrapolation of the wind or an iterative method is needed to obtain an accurate estimate of the trajectory. This paper proposes an alternative hybrid Semi-Lagrangian-Eulerian scheme whereby the semi-Lagrangian part of the trajectory and interpolation to departure points are performed explicitly and only once per time step with a known velocity field together with a relatively small implicit Eulerian correction. Unlike classical iterative SL schemes, the present method removes (SL) advection completely from the iterative process. We also show that the proposed scheme can also be used in several novel ways including: solving some singular problems associated with classical SL and accuracy issues near solid boundaries. The present approach can also be used as an efficient and straightforward splitting strategy.

Error growth and phase lag analysis for high Courant numbers

The type of numerical scheme used in numerical computation of fluid flow largely affects the accuracy of the solution. In order to obtain accurate numerical solution in a transient analysis, implicit schemes are known to be unconditionally stable for linear systems while non-linearity in advection systems often limits the stability. Nevertheless, the implicit schemes for non-linear systems allow larger Courant numbers than explicit schemes. However, these assertions have been addressed based on a simple numerical error growth analysis which does not account for dispersion and phase lag errors even in linear systems. Thus, motivated by the correct error dynamics given in Sengupta et al., (2007), it has been shown here that stable and accurate numerical solutions are possible for higher Courant numbers while their low values do not always lead to stable and accurate solutions. Here, higher Courant numbers can be achieved by finer grid spacing, keeping the time step same. Time integration is performed by implicit scheme, for which numerical stability is guaranteed and thus a higher Courant number implies higher resolution of the solution. This has been proven through a test case of the propagation of acoustic wave in water by considering the latter to be a compressible medium by solving unsteady Navier–Stokes equations. This observation is elucidated by evaluating the error growth with first order in time and second order in space schemes with the latter represented by fully implicit form at t = (n + 1)Δt. The results show that for high Courant numbers, the value of error growth is very close to unity and confirms to the needed neutral stability of the obtained numerical solution. Furthermore, the value of phase error nearly vanishes and reveals that numerical waves travel with nearly the same speed as physical waves.

Fully-coupled pressure-based algorithm for compressible flows: Linearisation and iterative solution strategies

The impact of different linearisation and iterative solution strategies for fully-coupled pressure-based algorithms for compressible flows at all speeds is studied, with the aim of elucidating their impact on the performance of the numerical algorithm. A fixed-coefficient linearisation and a Newton linearisation of the transient and advection terms of the governing nonlinear equations are compared, focusing on test-cases that feature acoustic, shock and expansion waves. The linearisation and iterative solution strategy applied to discretise and solve the nonlinear governing equations is found to have a significant influence on the performance and stability of the numerical algorithm. The Newton linearisation of the transient terms of the momentum and energy equations is shown to yield a significantly improved convergence of the iterative solution algorithm compared to a fixed-coefficient linearisation, while the linearisation of the advection terms leads to substantial differences in performance and stability at large Mach numbers and large Courant numbers. It is further shown that the consistent Newton linearisation of all transient and advection terms of the governing equations allows, in the context of coupled pressure-based algorithms, to eliminate all forms of underrelaxation and provides a clear performance benefit for flows in all Mach number regimes.

A New Numerical Model Coupling Modified Method of Characteristics and Galerkin Finite Element Method for Simulation of Solute Transport in Groundwater Flow System

A new numerical model coupling the modified method of characteristics (MMOC) with the Galerkin finite element method (GFE) is proposed for the assessment of groundwater pollution in regional groundwater flow system. The MMOC-GFE solves governing solute transport equation which involves the simulation of advection and dispersion parts by MMOC and GFE, respectively. This model allows the use of large time steps (large Courant numbers) and large spatial steps (large Peclet numbers) with stable and convergent solutions. The proposed model is successfully implemented for a test case which includes comparison of the model results with reported solutions of other numerical models. It is found that MMOC-GFE model is quite adequate in simulating solute transport in heterogeneous aquifer with combined pumping and injection schemes.

Comparison of dimensionally split and multi‐dimensional atmospheric transport schemes for long time steps

Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time steps can be achieved without significant cost using the flux‐form semi‐Lagrangian technique. The dimensionally split scheme used in this paper is constructed from the one‐dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi‐dimensional, method‐of‐lines scheme which, with implicit time‐stepping, is stable for Courant numbers significantly larger than 1.Two‐dimensional advection test cases on Cartesian planes are proposed which avoid the complexities of a spherical domain or multi‐panel meshes. These are solid‐body rotation, horizontal advection over orography and deformational flow. The test cases use distorted non‐orthogonal meshes either to represent sloping terrain or to mimic the distortions near cubed‐sphere edges.Mesh distortions are expected to accentuate the errors associated with dimension splitting, however the accuracy of the dimensionally split scheme decreases only a little in the presence of mesh distortions. The dimensionally split scheme also loses some accuracy when long time steps are used. The multi‐dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second‐order accuracy at high resolution. As is expected for implicit time‐stepping, phase errors occur when using long time steps but the spatially well‐resolved features are advected at the correct speed and the multi‐dimensional scheme is always stable.A naive estimate of computational cost (number of multiplies) reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi‐dimensional scheme is used instead with explicit time‐stepping, the Courant number is restricted to less than 1, the accuracy is maintained, and the cost becomes similar to the dimensionally split scheme.

Improving the convergence behaviour of a fixed‐point‐iteration solver for multiphase flow in porous media

SummaryA new method to admit large Courant numbers in the numerical simulation of multiphase flow is presented. The governing equations are discretized in time using an adaptive θ‐method. However, the use of implicit discretizations does not guarantee convergence of the nonlinear solver for large Courant numbers. In this work, a double‐fixed point iteration method with backtracking is presented, which improves both convergence and convergence rate. Moreover, acceleration techniques are presented to yield a more robust nonlinear solver with increased effective convergence rate. The new method reduces the computational effort by strengthening the coupling between saturation and velocity, obtaining an efficient backtracking parameter, using a modified version of Anderson's acceleration and adding vanishing artificial diffusion. © 2016 The Authors. International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd.

Contaminant transport at large Courant numbers using Markov matrices

Volatile organic compounds, particulate matter, airborne infectious disease, and harmful chemical or biological agents are examples of gaseous and particulate contaminants affecting human health in indoor environments. Fast and accurate methods are needed for detection, predictive transport, and contaminant source identification. Markov matrices have shown promise for these applications. However, current (Lagrangian and flux based) Markov methods are limited to small time steps and steady-flow fields. We extend the application of Markov matrices by developing a methodology based on Eulerian approaches. This allows construction of Markov matrices with time steps corresponding to very large Courant numbers. We generalize this framework for steady and transient flow fields with constant and time varying contaminant sources. We illustrate this methodology using three published flow fields. The Markov methods show excellent agreement with conventional PDE methods and are up to 100 times faster than the PDE methods. These methods show promise for developing real-time evacuation and containment strategies, demand response control and estimation of contaminant fields of potential harmful particulate or gaseous contaminants in the indoor environment.