Correcting the Semi-Implicit Numerical Scheme Incorporated in the KORSAR Code Two-Liquid Model

An efficient semi-implicit immersed boundary method for the Navier–Stokes equations

The combination of semi-implicit and semi-Lagrangian marching algorithms leads to stable integration of the meteorological equations with long time steps even for large advecting velocities and fast-moving free waves. In recent years, however, attention has been drawn to the fact that forced stationary waves may not be handled properly when the time step exceeds some limit dictated by the advecting velocity and the mesh size. The MC2 (Mesoscale Compressible Community) model is used to investigate numerically the behavior of internal gravity waves forced by stable flow over topography as a function of time step. Although the semi-implicit semi-Lagrangian scheme employed in the MC2 model allows larger time steps than classic Eulerian schemes, it leads to an inaccurate solution when the time step is increased beyond some limit. A stability analysis of the numerical response is conducted in order to explain model results. The response of two off-centered semi-implicit schemes proposed by Rivest et al...

A 6M digital twin for modeling and simulation in subsurface reservoirs

We have studied two numerical strategies for solving a coupled system of dictinct nonlinear equations governing sediment transport in Lake Okeechobee. Using high-resolution bathymetry data of Lake Okeechobee, Florida, we study the numerical properties of the two strategies, from 1 core to 512 cores. The fully-explicit scheme is straightforward to implement and requires a relatively small amount of computation per time step. However, this simple numerical strategy has to use small time steps to ensure stability. These small time steps may render the explicit solver impractical for long-term and high-resolution basin simulations. As a comparison, we have implemented a semi-implicit scheme, where the two partial differential equations at each time step are solved implicitly in sequence. Numerical experiments show that this semi-implicit scheme is numerically stable even for very large time steps. Using a multicore-based cluster, we have carried out parallel simulations of sediment transport along a river chanel and into Lake Okeechobee.

Reacting flow calculations are stiff for time-stepping. Recently, Zhong derived three sets of semi-implicit Runge-Kutta(SIRK) methods for split ordinary differential equations in the form of u' = /(«) 4g(u), where the nonstiff term / is treated explicitly while the stiff term g is simultaneously treated implicitly. The purpose of this work is twofold: first, to simulate multidimensional detonation waves using the high-order Rosenbrock semi-implicit Runge-Kutta time-stepping scheme(SIRK-3C); secondly to quantitatively compare the performance of SIRK with the second-order timesplitting schemes in dealing with the hyperbolic conservation laws with stiff relaxation terms. In the simulation of two-dimensional detonations, a full set of elementary hydrogen-oxygen kinetics with nine species are linear-implicitly treated while the explicit discretization of the basic advection equations are achieved by the third-order ENO schemes. Several model problems of hyperbolic systems with stiff relaxation are considered in the comparison of the Time-Splitting and the SIRK-3C scheme. A set of low-storage SIRK(LSSIRK) schemes is derived and tested to be high-order accurate and strongly A-stable. The results show that high-order SIRK time-stepping methods are suitable for stiff reactive flow simulations like the multi-dimensional detonations. INTRODUCTION In the reactive flow field solutions and the combustion related problems, the existence of several nonequilibrium states impose additional difficulties in the solutions of the reactive Euler equations with stiff relaxation terms. Since the smallest time scale is introduced from the chemical reaction kinetics, the second-order time-splitting method of Strang^ allows an independent calculation of the stiff ODE for the source term via implicit methods in one step, and an explicit highorder convective calculation in the following time steps. By the main source of stiffness from the basic hyperbolic conservation laws, it can accomplish a •Graduate Student, Member AIAA 'Assistant Professor, Mechanical and Aerospace Engineering Department, Member AIAA Copyright ©1997 by American Institute of Aeronautics and Astronautics, Inc. AH rights reserved. robust and easy-to-implement computation of reactive flow of current interests. Despite the second-order in time accuracy and the robustness of these methods, there are two main difficulties with the stiffness which have been minimally handled. The spurious solutions are one of the often observed phenomena, which are stable and free of oscillations, and yet may be completely incorrect.'-' Some type of resolution technique is needed to treat this known difficulties.^'!] The degrading of the second-order time accuracy to the first-order is another and quite recent observation when there exist solutions with many small-scaled structures of high-degree of complication. Jin® has modified the Strang splitting into the secondorder time-splitting methods which preserve its highorder accuracy when the considered solutions are complicated. New third-order semi-implicit Rosenbrock type Runge-Kutta scheme, based on the original version of Zhong'---', is computationally efficient with the linearized source term integration, thus requiring no iterations during the implicit treatment of the stiff source term. Third-order ENO schemes discretize the convective fluxes such that both implicit and explicit terms are treated at each of the three Runge-Kutta stages. This high-order Runge-Kutta scheme(SIRK-3C) still possesses the typical under-resolved characteristics of unphysical spurious numerical results of the TimeSplitting method. However the reduction to lower order if the small relaxation time is not temporally wellresolved, is successfully removed. Multi-dimensional Detonations Detonation waves are multi-dimensional and unstable phenomenon in nature as demonstrated by the early experiments of Urtiew and Oppenheinv-'. Existence of the triple points'^, consisting of an incident shock, a reflecting shock and a Mach stem, is the main characteristic of the reacting region behind the propagating shock front, and these detaching triple points from the leading front further contributes to the rolling up of vortices of opposite strength. One distinctive feature of the instability process observed in experiments is the formation of regular cell structures as triple points collide as they come together in the incident shock and move away from each other in the newly formed Mach stem. Figure[13] depicts this process of triple point col-

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

In molecular dynamics simulations, the limited time step size has been a barrier to simulating long-time behaviors. Implicit time integration methods allow markedly larger time steps than the standard explicit time method, although they have major drawbacks such as overheads solving linear systems and instability of Newton iterations. To overcome these issues, we propose a semi-implicit time integration scheme, the semi-implicit Hessian correction (SimHec) scheme, for overdamped Langevin dynamics. The method focuses on the Hessian matrices of bonded and nonbonded interactions, where components with large negative Hessian eigenvalues are cut off in the linear approximation of momentum equations to avoid instability. The narrow band Hessian matrix enables an efficient parallelized linear solution with an overlapping approximation. We tested SimHec for the interdomain fluctuations in adenylate kinase and the powerstroke transition of myosin II using a coarse-grained protein model. SimHec reproduced the same dynamics as the explicit method, although the transition dynamics tended to be accelerated and fluctuations in bonded potentials were slightly reduced. These deviations were corrected using a hybrid method, SimHec-H, which adds explicit time steps after the semi-implicit time step. The proposed scheme allowed us to use time steps 50-200 times larger than those in explicit time integration, which resulted in a speedup factor of 7-30 taking the overhead into account.

One of the principle features of RELAP5-based system thermal hydraulic codes is the use of a two-fluid, non-equilibrium, non-homogeneous, hydrodynamic model for the transient simulation of the two-phase system behavior. This model includes six governing equations to describe the mass, energy, and momentum of the two fluids. The current version of RELAP-5 uses non-conservative numerical approximation form of conservation equations. The current version of RELAP5 versions have mass and energy errors during time advancements, either resulting in (a) automatic reduction of time steps used in the advancement of the equations and increased run times or (b) the growth of unacceptably large errors in the transient results. Therefore, conservative conservation equations and closure equations were developed to address this problem in the first part of the paper series This part of the series demonstrates the numerical approach to implement the developed conservative conservation equations into RELAP5 and the results of RELAP5 including developed conservative form of conservation equations. RELAP5 versions including conservative and non-conservative conservation equations are compared for various tests from a single pipe to a whole Pressurized Water Reactor (PWR) model.

A mathematical model of simultaneous heat and moisture transfer is developed for convective drying of building material. A rectangular brick is considered for sample object. Finite-difference method with semi-implicit scheme is used for solving the transient governing heat and mass transfer equation. Convective boundary condition is used, as the product is exposed in hot air. The heat and mass transfer equations are coupled through diffusion coefficient which is assumed as the function of temperature of the product. Set of algebraic equations are generated through space and time discretization. The discretized algebraic equations are solved by Gauss–Siedel method via iteration. Grid and time independent studies are performed for finding the optimum number of nodal points and time steps respectively. A MATLAB computer code is developed to solve the heat and mass transfer equations simultaneously. Transient heat and mass transfer simulations are performed to find the temperature and moisture distribution inside the brick.

A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

A significant number of atmospheric models use a semi-implicit algorithm in order to carry out the time integration. In the semi-implicit scheme, the size of the time step is no longer restricted by the phase speed of external gravity waves; it is limited mainly by the magnitude of the wind associated with advection. Combining a semi-Lagrangian scheme with the semi-implicit technique introduces the possibility of integrating atmospheric models using relatively large time steps. In this paper, the semi-implicit algorithm in association with the semi-Lagrangian technique is presented. To solve the equations of semi-implicit schema we have used method of dimension reduction which reduces the three-dimensional problem to two-dimensional problems. The original contribution of this way is that semi-implicit approximation leads to the solution of two-dimensional separable partial differential equations. A detailed description is given and the model is tested by using the Lambert projection and the initial data provided by the ARPEGE model. The results are very satisfactory.

We consider the second order Strang splitting method to approximate the solution to the KdV equation. The model equation is split into three sets of initial value problems containing convection and dispersal terms separately. TVD MUSCL or MUSCL scheme is applied to approximate the convection term and the second order centered difference method to approximate the dispersal term. In time stepping, explicit third order Runge-Kutta method is used to the equation containing convection term and implicit Crank-Nicolson method to the equation containing dispersal term to reduce the CFL restriction. Several numerical examples of weakly and strongly dispersive problems, which produce solitons or dispersive shock waves, or may show instabilities of the solution, are presented.

Transient model is used to analyze the dynamic development processes of vacuum arc, which is necessary especially under the high frequency vacuum arc. In this paper, the transient magneto-hydrodynamic of vacuum arc is established. The commercial software FLUENT is used to solve mass conservation equation, momentum conservation equation, conservation equation of ion energy, conservation equation of electron energy and magnetic field transport equation. The distribution of ion pressure, ion velocity, ion temperature and electron temperature will be discussed under different frequency, current and materials. With the rising of frequency, the unsteady terms in the function increase obviously, which greatly affect the development processes of the vacuum arc. The maximum current that a small-size trigger tube can carry is limited by its size. The current mainly influences the joule heating and the boundary condition at cathode size. The increase of current leads to the increase of the distribution of ion pressure and temperature. The influence of different materials is mainly due to the difference of the ion mass. Larger relative atomic mass results in higher ion pressure and lower ion temperature and electron temperature under the same mean charge number.

This paper presents a new method of semianalytical particle tracking analysis under transient conditions. To permit accurate particle tracking in transient flows, the particle velocity is interpolated linearly in both space and time coordinates within each finite difference cell and time step. By using the total derivative and this interpolation scheme, the particle velocity is derived and integrated analytically to obtain the particle's trajectory within each cell and time step. Errors are much smaller than in numerical integration schemes. Numerical experiments show that the transient semianalytical solutions are efficient and accurate.

Asymptotically preserving particle methods for strongly magnetized plasmas in a torus