Semi Implicit Research Articles

Importance of temporal symmetry in spatial discretization for geostrophic adjustment in semi‐implicit Z‐grid schemes

Among the dynamical cores of numerical weather prediction communities, many different discretization methods can be distinguished to solve the equations governing the motions in the atmosphere numerically. One of them, the Z‐grid approach, is based on solving the equations formulated in terms of divergence and vorticity on an Arakawa A‐grid, a grid where all the variables are defined at the same grid points. To permit an efficient semi‐implicit (SI) treatment, Z‐grid schemes were proposed in the literature that first perform SI time discretization on the momentum equations formulated in terms of velocity components in order to construct from this a discretized divergence equation. This publication shows that a careful formulation of such SI Z‐grid schemes is required to conserve appropriate dispersion relations for inertia–gravity, inertia–Lamb and Rossby waves. It is proven analytically for a two time‐level (2TL) SI Z‐grid scheme of the 1D shallow‐water equations that the spatial discretization must respect temporal symmetry, meaning that the spatial discretization must be identical in the implicit and explicit parts of the scheme. If not, the discretized waves are damped or amplified and their phase and group velocity may be seriously distorted. These findings are discussed in detail and both 1D and 2D numerical tests are carried out to demonstrate that a symmetric formulation is an important modelling constraint in order to obtain an appropriate geostrophic adjustment.

Numerical Simulation of Unidirectional Stratified Flow by Moving Particle Semi Implicit Method

AbstractNumerical simulation of stratified flow of two fluids between two infinite parallel plates using the Moving Particle Semi-implicit (MPS) method is presented. The developing process from entrance to fully development flow is captured. In the simulation, the computational domain is represented by various types of particles. Governing equations are described based on particles and their interactions. Grids are not necessary in any calculation steps of the simulation. The particle number density is implicitly required to be constant to satisfy incompressibility. The weight function is used to describe the interaction between different particles. The particle is considered to constitute the free interface if the particle number density is below a set point. Results for various combinations of density, viscosity, mass flow rates, and distance between the two parallel plates are presented. The proposed procedure is validated using the derived exact solution and the earlier numerical results from the Level-Set method. Furthermore, the evolution of the interface in the developing region is captured and compares well with the derived exact solutions in the developed region.

Numerical simulation of free surface water wave for the flow around NACA 0015 hydrofoil using the volume of fluid (VOF) method

The surface wave generated by flow around NACA 0015 hydrofoil moving near free surface of water is simulated numerically in this study. The two-dimensional implicit finite volume method (FVM) is applied to solve Reynolds Averaged Navier–Stokes (RANS) equation. The realizable κ-ε turbulence model has been implemented to capture turbulent flow around the hydrofoil in the free surface zone at different submergence ratios (h/c). The volume of fluid (VOF) method has been used to determine the free surface effect of water. For pressure–velocity coupling, SIMPLEC (Semi IMPlicit Linked Equations Consistent) algorithm is employed. The second order upwind scheme is applied for discretization of momentum, volume fraction, turbulent kinetic energy and turbulent dissipation rate. At first NACA 0012 hydrofoil section is analyzed at h/c=0.91 and the result is validated by comparing with the published experimental result. Finally, the analysis is carried out with NACA 0015 hydrofoil section for seven h/c ratios, ranging from 0.91 to 4.0. The profile of the waves, the contours of velocity magnitude and static pressure near the hydrofoil and free surface, and the values of lift and drag coefficients are computed at Fn=0.5711, Re=1.592×105 for those submergence ratios.

An inherently mass‐conserving semi‐implicit semi‐Lagrangian discretization of the deep‐atmosphere global non‐hydrostatic equations

Following previous work on an inherently mass‐conserving semi‐implicit (SI) semi‐Lagrangian (SL) discretization of the two‐dimensional (2D) shallow‐water equations and 2D vertical slice equations, that approach is here extended to the 3D deep‐atmosphere, non‐hydrostatic global equations. As with the reduced‐dimension versions of this model, an advantage of the approach is that it preserves the same basic structure as a standard, non‐mass‐conserving, SISL version of the model. Additionally, the model is simply switchable to hydrostatic and/or shallow‐atmosphere forms. It is also designed to allow simple switching between various geometries (Cartesian, spherical, spheroidal). The resulting mass‐conserving model is applied to a standard set of test problems for such models in spherical geometry and compared with results from the standard SISL version of the model.

Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.

Numerical investigation of entropy generation in an unsteady flow through a porous pipe with suction

This paper investigates the entropy generation rates in an unsteady flow of a variable viscosity incompressible fluid through a porous pipe with uniform suction at the surface. The nonlinear governing equations for momentum and energy balance are derived and solved numerically using a semi–implicit finite difference scheme in order to obtain the fluid velocity and temperature profiles. Numerical results for volumetric entropy generation numbers, irreversibility distribution ratio and the Bejan number are presented graphically and discussed quantitatively for various values of the embedded parameters.

建築物に作用する津波遡上流による流体圧の評価法に関する研究

The authors propose the suitable way of simulation with the MPS method when run-up tsunami acts on architecture in this paper.1. The MPS method that is semi implicit method costs a lot of computational time. For this reason, to use the wave maker with the functionality of inflow-outflow for decreasing particle number lead to high-speed simulation.2. In the MPS method, detection accuracy of free surface is low. and it leads to pressure oscillation. The new way focusing on continuity of free-surface-particles is able to improve the accuracy of detecting free surface, and to reduce pressure oscillation.3. The authors reproduce Tsunami acting on an architecture with the tank experiment and the MPS method with the new way developed and the proposed method, and compare tsunami pressure acting on the front of it. As a result, the MPS method is able to simulate Tsunami acting on architecture including the impulsive pressure.

Contribution of offshore islands in the prediction of water levels due to tide–surge interaction for the coastal region of Bangladesh

In this study, a doubly nested tide–surge interaction model was established for the coastal region of Bangladesh. A fine grid model, capable of incorporating all major offshore islands, was nested into a coarse grid model extending up to 15°N latitude of the Bay of Bengal. To take into account the thickly populated small and big islands between Barisal and Chittagong and the extreme bending of the coastline accurately, a very fine grid model for this region was again nested into the fine grid model. Along the northeast corner of this very fine grid model, the Meghna River discharge was taken into account. The boundaries of the coast and islands were approximated through proper stair step, and the model equations were solved by semi–implicit finite difference technique using staggered grid. Appropriate tidal regime over the model domain was generated by forcing the sea level to be oscillatory with the constituent M2 along the southern open boundary of the coarse grid model omitting wind stress. This previously generated tidal regime was introduced as the initial state of the sea for nonlinear tide–surge interaction phenomenon. The model was applied to estimate water levels along the coastal region of Bangladesh due to the interaction of tide and surge associated with the storm April 1991, and the results were found to be in a reasonable agreement with those observed. The model was used to investigate the influence of offshore islands on water levels and water levels were found to be significantly influenced by offshore islands.

Laplace transform integration of the shallow‐water equations. Part II: Semi‐Lagrangian formulation and orographic resonance

AbstractIn this article we combine the Laplace transform (LT) scheme with a semi‐Lagrangian advection scheme, and implement it in a shallow‐water model. It is compared to a reference model using the semi‐implicit (SI) scheme, with both Eulerian and Lagrangian advection. We show that the LT scheme is accurate and computationally competitive with these reference schemes. We also show, both analytically and numerically, that the LT scheme is free from the problem of orographic resonance that is found with semi‐implicit schemes. Copyright © 2011 Royal Meteorological Society

Laplace transform integration of the shallow‐water equations. Part I: Eulerian formulation and Kelvin waves

AbstractA filtering integration scheme is developed, using a modification of the contour used to invert the Laplace transform (LT). It is shown to eliminate components with frequencies higher than a specified cut‐off value. Thus it is valuable for integrations of the equations governing atmospheric flow. The scheme is implemented in a shallow‐water model with an Eulerian treatment of advection. It is compared to a reference model using the semi‐implicit (SI) scheme. The LT scheme is shown to treat dynamically important Kelvin waves more accurately than the SI scheme. Copyright © 2011 Royal Meteorological Society

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

AbstractWe extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.

Updating preconditioners for nonlinear deblurring and denoising image restoration

Image restoration is the recovery of images that have been degraded by blur and noise. Nonlinear degenerate diffusion partial differential equation models for image restoration require often the solution of a challenging discrete problem. We consider solving the related discrete models in the time-scale steps by Krylov iterative solvers accelerated by updating a preconditioner based on incomplete factorizations which presents a global computational cost slightly more than linear in the number of the image pixels. We demonstrate the efficiency of the strategy by denoising and deblurring some images with a generalized Alvarez–Lions–Morel-like partial differential equation model discretized by a semi implicit complementary volume scheme.

An assessment of turbulence models for the prediction of flow past a circular cylinder with momentum injection

Reynolds averaged Navier–Stokes equations (RANS) are solved to simulate the flow past a circular cylinder. Momentum injection through Moving Surface Boundary-layer Control (MSBC) with zero net mass injection is implemented to achieve wake control. Popular eddy viscosity based closure models are assessed for their predictive capability of turbulent wake characteristics. For the present simulation, sub-critical Reynolds number ( Re) of 3900 is chosen, where extensive validations are available. A stabilization approach is proposed to model, predict and control vortex shedding behind a circular cylinder. Along with the mass, momentum conservation, turbulent kinetic energy (TKE) and its rate of dissipation equations are solved with the objective of achieving the annihilation of wake structures. To enable momentum injection, two simple rotary type control cylinders are located at 120°, behind the main cylinder. The ratio of the main cylinder, control cylinder and gap between them are fixed at D: 0.1D: 0.01D, respectively. These control cylinders, which are like externally controllable actuators, are assessed for their ability to influence momentum injection and hence wake patterns. The popular finite volume based Semi Implicit Pressure Linked Equations (SIMPLE) scheme is employed for the numerical calculations. Detailed assessment of different eddy viscosity based turbulence models viz., standard k – ε , Renormalization Group (RNG) k – ε , realizable k – ε and k – ε version of Kato–Launder (KL) is carried out. As a precursor, validation of turbulence statistics such as, mean streamwise velocity along the wake centerline, mean pressure coefficient on the cylinder surface and time averaged Reynolds stresses etc. is carried out against known experimental and numerical computations. The role of externally controllable actuators on the fluid flow patterns past a circular configuration is assessed with the help of streaklines, streamlines, vorticity, Reynolds stress contours etc. Complete suppression of vortex shedding is observed for an injection parameter (defined as the ratio of control cylinder velocity ( U c ) to that of the freestream ( U ∞ )) ξ = 6.0 . The results clearly demonstrate the effectiveness of a rather simple momentum injection strategy in suppressing turbulent vortex structures.

On a semi-implicit scheme for spectral simulations of dispersive magnetohydrodynamics

The effectiveness of a semi-implicit (SI) temporal scheme is discussed in the context of the dispersive magnetohydrodynamics where, due to the whistler modes, stability of explicit algorithms requires a time step decreasing quadratically as the resolution is linearly increased. After analyzing the effects of this scheme on the Alfvén-wave dispersion relation, spectral simulations of nonlinear initial value problems where small-scale dispersion has a main effect on the global dynamics are presented. Permitting a moderate, albeit significant, increase of the time step for a minor additional cost relatively to explicit schemes, the SI algorithm provides an efficient tool in situations, such as turbulent regimes, where the time steps making fully implicit schemes efficient are too large to ensure a satisfactory accuracy.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker–Planck equation

The Fokker–Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker–Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

A complete numerical model for electrokinetic flow and species transport in microchannels

A theoretical model is developed to investigate electrokinetic driven fluid flow and species transport. A typical cross-channel geometry is chosen with applications of species on-chip injection-separation analysis. The electrical potential, fluid flow and species concentration fields are presented in detail. These governing equations are non-dimensionalized uniquely to identify the controlling factors in microchannel performance. For the unsteady species transport equation, two different non-dimensionalization methods are compared. A preferred way to non-dimensionalize time in the concentration transport equation provides superior insight and understanding. Using a non-uniform staggered grid, a finite control volume method and line-by-line iterative procedure are implemented for the simulations. The Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is employed to solve the discrete equations. Grid independence and convergence studies are performed.

SIMPLE‐type preconditioners for the Oseen problem

AbstractIn this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.

An iterative semi-implicit scheme with robust damping

An efficient, iterative semi-implicit (SI) numerical method for the time integration of stiff wave systems is presented. Physics-based assumptions are used to derive a convergent iterative formulation of the SI scheme which enables the monitoring and control of the error introduced by the SI operator. This iteration essentially turns a semi-implicit method into a fully implicit method. Accuracy, rather than stability, determines the timestep. The scheme is second order accurate and shown to be equivalent to a simple preconditioning method. We show how the diffusion operators can be handled so as to yield the property of robust damping, i.e. dissipating the solution at all values of the parameter DΔt, where D is a diffusion operator and Δt the timestep. The overall scheme remains second order accurate even if the advection and diffusion operators do not commute. In the limit of no physical dissipation, and for a linear test wave problem, the method is shown to be symplectic. The method is tested on the problem of Kinetic Alfvén wave mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is used. A two-field gyrofluid model is used and an efficacious k-space SI operator for this problem is demonstrated. CPU speed-up factors over a CFL-limited explicit algorithm ranging from ∼20 to several hundreds are obtained, while accurately capturing the results of an explicit integration. Possible extension of these results to a real-space (grid) discretization is discussed.

Improvement of a 2-D SIA ice-flow model: application to Glacier de Saint-Sorlin, France

A number of improvements have been made to an existing two-dimensional ice-flow model applied to an alpine glacier. Analysis of the results of the existing model revealed several shortcomings. The first concerns the lack of mass conservation of the applied alternating-direction-implicit (ADI) scheme. A semi-implicit (SI) scheme is therefore proposed and the effects on mass conservation assessed by a comparison with the ADI scheme. The comparison is first carried out with a simple theoretical glacier for which the improvement is significant. Concerning the real case of Glacier de Saint-Sorlin, France, the initial deviation in mass conservation was much less pronounced such that the new scheme, although improving mass conservation, does not significantly change the modelled dynamics. However, other shortcomings that have a more profound impact on the modelling of glacier behaviour have been identified. The ice thickness may become negative over some gridpoints, leading to an inconsistency. The problem is partly resolved by incorporating extra checks on critical gridpoints at the glacier border. Finally, with the help of ice particle tracking, unrealistic ice settlement above the bergschrund has been identified as the main reason for spurious dynamic effects and has been corrected.

Numerical simulation of temperature and concentration distributions in fluidized beds with liquid injection

In this paper a numerical simulation study of dynamic behavior of a fluidized bed with liquid injection is presented. A continuum model has been developed taking into account the mass and energy balances of solid, gas as well as liquid to describe the temperature and concentration distributions in gas–solid-fluidized beds. The model considers the deposition efficiency of the liquid droplets as well as the influence of the spray nozzle region. For solving the non-linear partial differential equations with discrete boundary conditions a finite element method is used. Numerical computations have been done with two different schemes of time integration, a fully implicit and a semi implicit scheme. The complex correlations of mass and liquid flow rates, mass and heat transfer, drying, and transient two-dimensional air humidity, air temperature, particle wetting, liquid film temperature and particle temperature were simulated. The model was validated with transient measurements of the air temperature and air humidity at the outlet of a fluidized bed with water injection.