Courant-Friedrichs-Lewy Research Articles

Acoustic finite-difference modeling beyond conventional Courant-Friedrichs-Lewy stability limit: Approach based on variable-length temporal and spatial operators

Conventional finite-difference (FD) methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy (CFL) numbers 0.707 and 0.577 for two-dimensional (2D) and three-dimensional (3D) equal spacing cases, respectively, thereby limiting time step selection. Based on the definition of temporal and spatial FD operators, we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy. First, to simulate wave propagation beyond the conventional CFL stability limit, the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones. Second, to preserve the modeling accuracy, the velocity-dependent lengths of the temporal and spatial operators are adaptively varied. The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples, respectively. We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.

Multiscale Numerical Modeling of Solute Transport with Two-Phase Flow in a Porous Cavity

This paper introduces dimensional and numerical investigation of the problem of solute transport within the two-phase flow in a porous cavity. The model consists of momentum equations (Darcy’s law), mass (saturation) equation, and solute transport equation. The cavity boundaries are constituted by mixed Dirichlet-Neumann boundary conditions. The governing equations have been converted into a dimensionless form such that a group of dimensionless physical numbers appear including Lewis, Reynolds, Bond, capillary, and Darcy numbers. A time-splitting multiscale scheme has been developed to treat the time derivative discretization. Also, we use the Courant-Friedrichs-Lewy (CFL) stability condition to adapt the time step size. The pressure is calculated implicitly by coupling Darcy’s law and the continuity equation, then, the concentration equation is solved implicitly. Numerical experiments have been conducted and the effects of the dimensionless numbers have been on the saturation, concentration, pressure, velocity, and Sherwood number have been investigated.

An implicit kinetic inviscid flux for predicting continuum flows in all speed regimes

In this study, the kinetic inviscid flux (KIF) is improved and coupled with an implicit strategy. The KIF is a recently proposed numerical method, which is a dynamic combination of the kinetic flux vector splitting (KFVS) method and the totally thermalized transport (TTT) method. The inherent microscopic mechanism of the KFVS makes the KIF good at solving shock waves and avoiding the numerical shock instability phenomenon. When developing the implicit KIF, it is noticed that, in boundary layers, the KFVS part of the KIF not only reduces the accuracy but also seriously reduces the Courant–Friedrichs–Lewy (CFL) number. As a result, a new weight is proposed in this paper to combine the KFVS method with the TTT method properly. Besides admitting the use of larger CFL numbers, this new weight also contributes to more accurate numerical results like pressure, friction coefficient, and heat flux when solving shock waves, boundary layers, and complex supersonic/hypersonic flows. To examine the validity, accuracy, and efficiency of the proposed method, six numerical test cases covering the whole speed regime are conducted, including the hypersonic viscous flow past a cylinder, the hypersonic double-cone flow, the hypersonic double-ellipsoid flow, the laminar shock-boundary layer interaction, the supersonic flow around a ramp segment and the subsonic lid-driven cavity flow.

On explicit discontinuous Galerkin methods for conservation laws

The concept of projection applied to explicit discontinuous Galerkin (DG) schemes is investigated. The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz called space–time expansion discontinuous Galerkin or STE-DG scheme (Lörcher et al. 2007, Gassner et al. 2008), and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction takes place; the two methods differ, however, in the evaluation of the projections in the space-time volume integrals. The STE-DG scheme evaluates these in a straightforward manner, whereas the moment scheme employs a successive procedure with each moment update uses the results by the lower-order updates. The trade-off is that the moment scheme has the disadvantage of a more elaborate corrector step and the significant advantage of a CFL (Courant-Friedrichs-Lewy) condition of 1 for all (degree) p and accuracy order of 2p+1 (super accuracy property) for one-dimensional (1D) advection. In contrast, the STE-DG method is accurate to only the expected order of p+1 and has a more restrictive CFL condition. Due to the predictor-corrector formulation that does not involve methods of characteristics, these schemes extend easily to systems of equations in multiple dimensions. Concerning the case of two spatial dimensions (2D), for an advection using a Cartesian grid, when the flow does not align with the axes, especially when it is along a diagonal direction, the CFL conditions for the moment schemes also become restrictive and need improvement as will be shown by Fourier (von Neumann) analyses.

Further Assessment of Three Bathe Algorithms and Implementations for Wave Propagation Problems

This paper further analyzes three Bathe algorithms ([Formula: see text]-Bathe, [Formula: see text]-Bathe and [Formula: see text]-Bathe) with their unknown properties revealed. The analysis shows firstly that three Bathe algorithms can cover two common integration schemes, trapezoidal rule and backward Euler formula, and that the second-order [Formula: see text]-Bathe algorithm is algebraically identical to the [Formula: see text]-Bathe algorithm. Via formulation of the generalized two-sub-step Newmark algorithm, it is shown that the common Newmark method cannot be considered as a special case of the [Formula: see text]-Bathe algorithm. For wave propagation problems, optimal Courant–Friedrichs–Lewy (CFL) numbers for reducing dispersion errors are found for the three Bathe algorithms by considering spatial and temporal discretizations simultaneously, while the modified integration rules are used for the element mass and stiffness matrices to reduce the anisotropy in wave propagating directions. The recommended optimal algorithmic parameters are given for the three Bathe algorithms to help users effectively solve various dynamic and wave propagation problems.

A ghost-cell immersed boundary method for unified simulations of flow over finite- and zero-thickness moving bodies at large CFL numbers

A ghost-cell immersed boundary method for unified simulations of flow over finite- and zero-thickness moving bodies at large Courant-Friedrichs-Lewy (CFL) numbers is presented. In order to handle such bodies in a unified manner, algorithms for interface construction and cell demarcation are proposed. The main challenge in treating zero-thickness bodies is to maintain sharpness and accuracy even at large CFL numbers with diminished spurious force oscillations. Thus, the effect of large CFL numbers on the solution accuracy of fluid-structure interaction (FSI) problems involving zero-thickness bodies is investigated and necessary treatments to preserve solution accuracy even at large CFL numbers are suggested. The present study suggests two treatments which are important in preserving the accuracy and stability of the solution: backward time integration for computational cells called ‘swept-cells’ and pressure boundary condition with mass conservation. Composite implicit time integration for the dynamic equation of a thin elastic structure is employed for a stable simulation of FSI at large CFL numbers. By using large time step sizes, the present method not only enhances computational efficiency, but also suppresses spurious force oscillations while maintaining the sharpness of an infinitesimally thin body. The efficacy and accuracy of the present method are examined through numerical examples.

High-Accurate Non-Uniform Grids for System-Combined ADI-FDTD Method in Near-Field Scattering With Proper CFL Factor

In this paper, a high-accurate technique with non-uniform grids is introduced into a system-combined alternative-direction-implicit finite-difference time-domain (SC-ADI-FDTD) algorithm, and then successfully used to analyze electromagnetic propagations. To our knowledge, the conventional FDTD with non-uniform grids can be effectively deal with some edges of the three-dimensional cubes and complicated structures of the tiny objects by modulating the local grid scales, which to the extent improves its reliability and accuracy. However, due to existing the finer grids in the local computational region and the inevitable Courant-Friedrichs-Lewy (CFL) limit in the conventional FDTD, the temporal interval must be determined by the minimum fine spatial grid, resulting in much larger temporal sampling density required during the whole computation process. As the advantage of circumventing the repeated variables, the non-uniform SC-ADI-FDTD (NUSC-ADI-FDTD) cannot only break through the CFL limit to implement the high-efficient computation, but also further save more CPU time in the local microstructure cases. Furthermore, the empirical formula between the spatial sampling density and the CFL factors can be obtained from the numerical fitting method after errors analysis. The numerical simulations of the electromagnetic scattering have been executed to illustrate feasibility and validity of our proposed method.

Simulation strategy of the full-scale ship resistance and propulsion performance

The ship powering performance can be predicted based on model tests or simulations. However, differences exist between the full-scale ship performance and extrapolated model data. Therefore, research on full-scale ship performance simulation has important scientific significance and engineering value. This study used the REGAL general cargo vessel to perform full-scale ship resistance and self-propulsion simulations for various grid numbers, time step sizes, and wall Y+ values and compared the calculation and empirical results. The resistance components, waveform, Courant–Friedrichs Lewy (CFL) number, pressure field, and wake flow field of various resistance simulation conditions were analysed. The wall Y+ value was observed to affect the capture of boundary layer flow, ship pressure field evolution, and waveform, which subsequently affect the ship resistance. A Y+ value within 200 was considered to be appropriate when simulating the powering performance of a full-scale ship. The waveform evolution was significantly affected by the time step size, which should meet the requirement that the CFL number on the free surface is less than 1. When simulating the self-propulsion of a full-scale ship based on sliding mesh, the CFL number on the interface should also be less than 1. Additionally, to obtain the correct propeller excitation force, it was recommended that the time step size be maintained below 1°/step (0.000076 LPP/v).

On the discretization of river networks for large scale hydrologic-hydrodynamic models

ABSTRACT The discretization of river networks is a critical step for computing flow routing in hydrological models. However, when it comes to more complex hydrologic-hydrodynamic models, adaptations in the spatial representation of model calculation units are further required to allow cost-effective simulations, especially for large scale applications. The objective of this paper is to assess the impacts of river discretization on simulated discharge, water levels and numerical stability of a catchment-based hydrologic-hydrodynamic model, using a fixed river length (Δx) segmentation method. The case study was the Purus river basin, a sub-basin of the Amazon, which covers an area that accounts for rapid response upstream reaches to downstream floodplain rivers. Results indicate that the maximum and minimum discharges are less affected by the adopted Δx (reach-length), whereas water levels are more influenced by this selection. It is showed that for the explicit local inertial one-dimensional routing, Δx and the α parameter of CFL (Courant-Friedrichs-Lewy) condition must be carefully chosen to avoid mass balance errors. Additionally, a simple Froude number-based flow limiter to avoid numerical issues is proposed and tested.

Local high-order conservative space-time adaptive mesh refinement with the Runge-Kutta time integration for solving the systems of conservation laws

Numerical simulation is widely used as a main way of researching the processes described by the systems of partial differential equations in the divergence form, like fluid dynamics, traffic flow, wave propagation. To obtain desired accuracy and resolution, the high-order numerical methods are used. In this paper we consider the Runge-Kutta (RK) time discretization, when the space discretization can be high-order finite volume method, discontinuous Galerkin method or others. Presence of discrete analogue of conservation is considered to be very important for such methods as it corresponds to the essential feature of simulated process, becoming especially vital at the discontinuous solutions. The adaptive mesh refinement (AMR) techniques are used to get the numerical solution with more resolution where needed. For the explicit numerical scheme obeying the Courant-Friedrichs-Lewy (CFL) condition, the bigger time integration step can be chosen for the coarser cells. The problem is how to reformulate the numerical scheme near the boundary between cells of different size, keeping the conservation and high-order accuracy. In this paper a new algorithm is suggested. The changes in the numerical method are made only in the vicinity of the boundary. The local character of the developed algorithm allows us to efficiently implement it at modern computer systems. Results of validation tests of developed solver are shown.

A framework to evaluate IMEX schemes for atmospheric models

Abstract. We present a new evaluation framework for implicit and explicit (IMEX) Runge–Kutta time-stepping schemes. The new framework uses a linearized nonhydrostatic system of normal modes. We utilize the framework to investigate the stability of IMEX methods and their dispersion and dissipation of gravity, Rossby, and acoustic waves. We test the new framework on a variety of IMEX schemes and use it to develop and analyze a set of second-order low-storage IMEX Runge–Kutta methods with a high Courant–Friedrichs–Lewy (CFL) number. We show that the new framework is more selective than the 2-D acoustic system previously used in the literature. Schemes that are stable for the 2-D acoustic system are not stable for the system of normal modes.

Coupling LES with soot model for the study of soot volume fraction in a turbulent diffusion jet flames at various Reynolds number configurations

PurposeThis study aims to investigate the insights of soot formation such as rate of soot coagulation, rate of soot nucleation, rate of soot surface growth and soot surface oxidation in ethylene/hydrogen/nitrogen diffusion jet flame at standard atmospheric conditions, which is very challenging to capture even with highly sophisticated measuring systems such as Laser Induced Incandescence and Planar laser-induced fluorescence. The study also aims to investigate the volume of soot in the flame using soot volume fraction and to understand the global correlation effect in the formation of soot in ethylene/hydrogen/nitrogen diffusion jet flame.Design/methodology/approachA large eddy simulation (LES) was performed using box filtered subgrid-scale tensor. A filtered and residual component of the governing equations such as continuity, momentum, energy and species are resolved and modeled, respectively. All the filtered and residual components are numerically solved using the ILU method by considering PISO pressure–velocity solver. All the hyperbolic flux uses the QUICK algorithm, and an elliptic flux uses SOU to evaluate face values. In all the cases, Courant–Friedrichs–Lewy (CFL) conditions are maintained unity.FindingsThe findings are as follows: soot volume fraction (SVF) as a function of a flame-normalized length for three different Reynolds number configurations (Re = 15,000, Re = 8,000 and Re = 5,000) using LES; soot gas phase and particulate phase insights such as rate of soot nucleation, rate of soot coagulation, rate of soot surface growth and soot surface oxidation for three different Reynolds number configurations (Re = 15,000, Re = 8,000 and Re = 5,000); and soot global correction using total soot volume in the flame volume as a function of Reynolds number and Froude number.Originality/valueThe originality of this study includes the following: coupling LES turbulent model with chemical equilibrium diffusion combustion conjunction with semi-empirical Brookes Moss Hall (BMH) soot model by choosing C6H6 as a soot precursor kinetic pathway; insights of soot formations such as rate of soot nucleation, soot coagulation rate, soot surface growth rate and soot oxidation rate for ethylene/hydrogen/nitrogen co-flow flame; and SVF and its insights study for three inlet fuel port configurations having the three different Reynolds number (Re = 15,000, Re = 8,000 and Re = 5,000).

An L1 Legendre–Galerkin spectral method with fast algorithm for the two-dimensional nonlinear coupled time fractional Schrödinger equation and its parameter estimation

In this paper, we derive an L1 Legendre–Galerkin spectral method with fast algorithm based on an efficient sum-of-exponentials (SOE) approximation for the kernel t−1−α to solve the two-dimensional nonlinear coupled time fractional Schrödinger equations. The numerical method is stable without the Courant–Friedrichs–Lewy (CFL) conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. At the same time, we use the fully implicit method to deal with the nonlinear terms. For the non-smooth solution, we employ the graded mesh method. Moreover, we estimate the parameters including fractional derivative index and the coefficients of the nonlinear term in the equations by applying the Cuckoo Search algorithm related to Lévy flights. Numerical examples are given to verify the theoretical analysis and confirm the effectiveness of the fast algorithm and the estimation method.

Data-Driven Model Reduction for Stochastic Burgers Equations.

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

Convergence acceleration for subiterative DDADI/D3ADI using multiblock implicit boundary condition

Developing accurate, robust and efficient implicit time-marching method is crucial for the rapid convergence of high Reynolds number wall-bounded flows. In the presence that a massive structured grid is partitioned into multiple multiblocks for parallelization, a real-time implicit boundary condition treatment taking the multiblock communications into account affects the convergence acceleration dramatically. In this paper, an implicit boundary condition (IBC) incorporated with the subiteration technique is proposed for DDADI and diagonalized DDADI (D3ADI) time-integration schemes to speed up their convergence rates. The compressible Reynolds-averaged Navier-Stokes (N–S) equations along with the Spalart-Allmaras (S–A) turbulence model can be advanced robustly in time with large admissible Courant–Friedrichs–Lewy (CFL) numbers. The accelerated efficiency stems from two reasons: (1) the subiteration technique is beneficial to eliminate the errors arising from the approximation factorization decomposition process; (2) the IBC technique is advantageous to remove the error accumulated at multiblock interfaces. The combination of these two techniques results in the fact that the implicit matrix equations are solved in an intact implicit manner no matter how many multiblock interfaces are involved. Four numerical cases are investigated where detailed comparisons of convergence acceleration are presented. The advantage of implicit boundary condition (IBC) over explicit boundary condition (EBC) is way more prominent. Meanwhile, the importance of keeping the implicit treatment of turbulence model to be consistent with N–S equations is also put forward, otherwise the convergence characteristics of N–S equations will be adversely affected. Lastly, with increasingly partitioned multiblocks, the subiterative DDADI/D3ADI with IBC still preserves good convergence behavior whereas those with EBC are likely to suffer from divergence.

Numerical simulation of impulse wave generation by idealized landslides with OpenFOAM

Landslide tsunamis and impulse waves are hazardous events with severe socioeconomic impacts. A long standing problem with simulations of these events is the generation stage, where landslides and water interact. Depth-averaged models like the Saint-Venant or Boussinesq Equations lose their validity for such applications. Therefore, we have to rely on a full treatment of the hydrodynamics, for instance by applying the Navier-Stokes Equations and Computational Fluid Dynamics (CFD). However, applications of fully three-dimensional methods to landslide tsunamis are sparse, and have often been outperformed by depth averaged models when compared to experimental data. In this work, we evaluate the multiphase Navier-Stokes Equations as implemented in OpenFOAM® in terms of impulse wave generation. We focus on a simplified two-dimensional setup where the landslide consists of water, in order to circumvent additional complexities due to treatment of landslide rheologies. We conduct a thorough grid refinement study and compare results to experiments to investigate model convergence, stability, and accuracy. The simulations display good agreement with the experimental data if the Courant-Friedrichs-Lewy (CFL) condition is modified to account for the specific properties of the multiphase system. Further, we use the validated model for sensitivity studies and to review various scaling relations for landslide generated tsunamis. The application of numerical models allows us to perform broad parametric tests and dissect the underlying physics of these predictive equations systematically. We found that the first wave crest may be well estimated by solely the landslide mass in our setting. Including additional properties related to landslide momentum can improve the predictive skill, while other parameters lead to no substantial improvement.

Modelling of 3D viscous fingering: Influence of the mesh on coreflood experiments

The modeling of Viscous Fingering (VF) is of interest to the study of multiphase flow in porous media, especially in the O&G industry, where this phenomenon is one of the root causes of high-water production and low sweep efficiency. In this study, 3D coreflood experiments found in the literature were simulated with CFD, and nine different structured and unstructured meshes were compared to determine which one gives the best description of this phenomenon. The meshes included the structured orthogonal mesh, commonly used in reservoir simulation, up to unstructured hexahedral, polyhedral, and tetrahedral meshes, among others. On the other hand, the CFD model, which was based on the Volume of Fluid (VOF) model and the porous media model, was validated against experimental data for the oil recovery profile with an absolute error below 7% for the best cases. It was found that VF is severely affected by the type of mesh and the Courant–Friedrichs–Lewy (CFL) number used. It was concluded that the unstructured polyhedral and structured hexahedral meshes gave the most reliable description of the finger growth dynamics and oil recovery compared to experimental data. Finally, a mesh independence test using the Grid Convergence Index (GCI) methodology, along with a CFL number analysis, allowed to determine that the modeling of this phenomenon should be made using an average CFL number of 0.25.

Auto Segmentation Method for Electromagnetic Radiation Camera Myocardial Images

Electromagnetic radiation camera (ERC) provides functional information about the left ventricle. We need new methods of quantitative analysis, such as the segmentation myocardial images for maximum utilization of ERC. The level set method(semi-automatic) based approach method that has excellent potential with regarding mapping topological change in 2D images but the crucial point is when stopping iteration to obtain the optimal segmentation result. This study suggests the Courant–Friedrichs–Lewy (CFL) condition, feature measurement index (FMI), and modified Thompson Tau technique (MTTT) to investigate the optimal convergence problem. The CFL condition contributes to the stable numerical iteration of the level set, whereas the FMI and MTTT are used for exact iteration stopping. This study compared the auto segmentation method and the manual segmentation method through assessments. The results were so matched (82.39±13.92 %) so we can develop a 3D model that could be used to diagnose ischemic heart disease and infarction. The proposed segmentation method can help to diagnose the disease conveniently and accurately in the clinic.

A Hybrid CN-FDTD-SPICE Solver for Field-Circuit Analyses in Low-Frequency Wideband Problems

The conventional finite-difference time-domain (FDTD) method is efficient for microwave frequencies, but requires a large number of time steps for low frequencies. In this work, a hybrid Crank–Nicolson (CN) FDTD-SPICE method is applied to field-circuit analyses in low- frequency wideband problems. The unconditionally stable CN-FDTD method is used to give a rapid full wave solution by overcoming the Courant–Friedrichs–Lewy (CFL) condition that constrains the conventional FDTD method at low frequencies. The hybrid CN-FDTD-SPICE method can be used for simulating the interactions between modern electronic circuit systems and a 3-D structure supporting field radiation and interferences. It can greatly improve the computational efficiency over the conventional FDTD method at low frequencies because its time increment can be significantly larger. Three numerical examples are presented to verify the accuracy and efficiency of this hybrid method. Its CPU time is much shorter than that of the conventional FDTD-SPICE method. This CN-FDTD-SPICE method becomes increasingly advantageous when the radiation fields are required as the system integration continues to accelerate.

An Implicit–Explicit Vertical Transport Scheme for Convection-Allowing Models

Abstract An adaptive implicit–explicit vertical transport method is implemented in the Advanced Research version of the Weather Research and Forecasting Model (WRF-ARW), and improved integration efficiency is demonstrated for configurations employing convective-allowing horizontal and vertical resolutions. During the warm season over the continental United States, stable forecasts at convective-allowing resolutions are more challenging because localized regions of extreme thermodynamic instability generate large vertical velocities within thunderstorms that cause the integrations to become unstable because of violations of the Courant–Friedrichs–Lewy (CFL) condition for the explicit advection scheme used in WRF-ARW. The implicit–explicit vertical transport scheme removes the CFL instability but maintains accuracy for typical vertical velocities. Tests using this scheme show that the new scheme permits a time step that is 20%–25% percent larger, and it reduces the wall clock time by 10%–13% percent relative to a configuration similar to a current operational convection-allowing model while also producing more realistic updraft intensities within the most intense storms. Other approaches to maintain stability are either less efficient (e.g., reducing the time step) or significantly impact the solution accuracy (e.g., increasing the damping and/or reducing the latent heating, which severely limits the updraft magnitudes during the forecasts).