Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics
equations

Abstract The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.

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Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽

10.3390/geosciences9010029 ◽

2019 ◽

Vol 9 (1) ◽

pp. 29 ◽

Cited By ~ 1

Author(s):

Paul M. Delgado ◽

V. M. Krushnarao Kotteda ◽

Vinod Kumar

Keyword(s):

Fixed Point ◽

Asymptotic Solution ◽

Relative Error ◽

Operator Splitting ◽

Splitting Method ◽

Approximate Solutions ◽

Time Step ◽

Operator Splitting Method ◽

The Stability ◽

Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

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A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2007-0014 ◽

2007 ◽

Vol 7 (3) ◽

pp. 227-238 ◽

Cited By ~ 23

Author(s):

S.H. Razavi ◽

A. Abolmaali ◽

M. Ghassemieh

Keyword(s):

Fourth Order ◽

Initial Conditions ◽

Time Integration ◽

Critical Time ◽

Linear Acceleration ◽

Time Step ◽

Time Integration Method ◽

Acceleration Method ◽

The Stability ◽

Critical Time Step

AbstractIn the proposed method, the variation of displacement in each time step is assumed to be a fourth order polynomial in time and its five unknown coefficients are calculated based on: two initial conditions from the previous time step; satisfying the equation of motion at both ends of the time step; and the zero weighted residual within the time step. This method is non-dissipative and its dispersion is considerably less than in other popular methods. The stability of the method shows that the critical time step is more than twice of that for the linear acceleration method and its convergence is of fourth order.

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Investigation Into Stability and Accuracy in Predicting Slender Bodies’ Hydroelasticity Using Loose Coupling Methods

Volume 6: Ocean Space Utilization ◽

10.1115/omae2015-41136 ◽

2015 ◽

Author(s):

Leixin Ma ◽

Shixiao Fu ◽

Ke Hu ◽

Qian Shi ◽

Runpei Li

Keyword(s):

Time Integration ◽

Hydrodynamic Forces ◽

Mass Force ◽

Loose Coupling ◽

Time Step ◽

Time Saving ◽

Coupling Methods ◽

The Stability ◽

Stability And Accuracy ◽

Hydroelastic Response

Problems concerning fluid-structure-interaction are often encountered in aquaculture engineering. For a moving slender structure like fishing net or floater in currents and waves, modified Morison Equation is a widely employed formula to estimate its hydrodynamic loads. The hydrodynamic forces are closely dependent on the structures’ velocity and acceleration, and quadratic relative velocity in the equation even adds nonlinearity in the forces. To study the hydroelastic response, two time-saving loosely coupling methods, calculating the hydrodynamic forces based on the structure’s response in the previous time step without iteration, are proposed in this paper. The loose coupling methods were proved to affect the traditional stability criteria for time integration. Based on the two loose coupling methods, the stability and accuracy of a slender beam’s hydroelasticity undergoing large deformation were studied. The calculated responses were compared against strong coupling results. It was found that if loose coupling is assumed in added mass force, unconditional instability is likely to occur. On the other hand, the accuracy of numerical results can be improved with smaller time increments set if loose coupling is only assumed in the quadratic relative drag force.

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Second Order Errors Related to Geometric Nonlinearity in Explicit Central Difference Operator

Volume 2: Computer Applications/Technology and Bolted Joints ◽

10.1115/pvp2008-61618 ◽

2008 ◽

Author(s):

Don R. Metzger ◽

Young-Suk Kim

Keyword(s):

Difference Method ◽

Difference Operator ◽

Time Integration ◽

General Nature ◽

Time Step ◽

Central Difference ◽

Central Difference Method ◽

Dynamic Structures ◽

The Stability ◽

Stability And Accuracy

Numerical analysis of nonlinear dynamic structures frequently makes use of the central difference method to step the transient forward in time. The method is particularly robust, accommodating material and geometric nonlinearities as well as contact surfaces and constraints of a very general nature. The implementation of the method is most usually performed according to [1], where velocity terms (or more generally rate quantities) are taken half a time step from the displacement and acceleration terms. It was recognized that a proper check of energy balance, requires that velocity must also be interpolated to the integer steps [2]. The stability and accuracy of the central difference method is well established, and decades of experience including its use in numerous commercial finite element codes confirms why it is the method of choice for explicit time integration of transients.

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A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽

10.2298/fil1715933k ◽

2017 ◽

Vol 31 (15) ◽

pp. 4933-4944

Author(s):

Dongseung Kang ◽

Heejeong Koh

Keyword(s):

Functional Equation ◽

Fixed Point ◽

General Solution ◽

Single Case ◽

Fixed Point Method ◽

Point Method ◽

Fixed Point Approach ◽

The Stability ◽

The Given ◽

Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

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A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽

10.3390/w12061639 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1639

Author(s):

Abdelkrim Aharmouch ◽

Brahim Amaziane ◽

Mustapha El Ossmani ◽

Khadija Talali

Keyword(s):

Finite Volume ◽

Coastal Aquifers ◽

Nonlinear Partial Differential Equations ◽

Time Integration ◽

Mass Conservation ◽

Coupled System ◽

Finite Volume Scheme ◽

Time Step ◽

Volume Scheme ◽

Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

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On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽

10.3390/math9010078 ◽

2020 ◽

Vol 9 (1) ◽

pp. 78

Author(s):

Haifa Bin Jebreen ◽

Fairouz Tchier

Keyword(s):

Differential Equation ◽

Galerkin Method ◽

Linear Ordinary Differential Equation ◽

Time Interval ◽

Hyperbolic Partial Differential Equation ◽

Time Step ◽

Numerical Examples ◽

One Dimensional ◽

The Stability ◽

The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

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Statistical Uncertainty of DNS in Geometries without Homogeneous Directions

Applied Sciences ◽

10.3390/app11041399 ◽

2021 ◽

Vol 11 (4) ◽

pp. 1399

Author(s):

Jure Oder ◽

Cédric Flageul ◽

Iztok Tiselj

Keyword(s):

Channel Flow ◽

A Priori ◽

Time Integration ◽

Navier Stokes ◽

Statistical Uncertainty ◽

Time Step ◽

Order Of Magnitude ◽

Averaging Time ◽

The Individual ◽

Time Required

In this paper, we present uncertainties of statistical quantities of direct numerical simulations (DNS) with small numerical errors. The uncertainties are analysed for channel flow and a flow separation case in a confined backward facing step (BFS) geometry. The infinite channel flow case has two homogeneous directions and this is usually exploited to speed-up the convergence of the results. As we show, such a procedure reduces statistical uncertainties of the results by up to an order of magnitude. This effect is strongest in the near wall regions. In the case of flow over a confined BFS, there are no such directions and thus very long integration times are required. The individual statistical quantities converge with the square root of time integration so, in order to improve the uncertainty by a factor of two, the simulation has to be prolonged by a factor of four. We provide an estimator that can be used to evaluate a priori the DNS relative statistical uncertainties from results obtained with a Reynolds Averaged Navier Stokes simulation. In the DNS, the estimator can be used to predict the averaging time and with it the simulation time required to achieve a certain relative statistical uncertainty of results. For accurate evaluation of averages and their uncertainties, it is not required to use every time step of the DNS. We observe that statistical uncertainty of the results is uninfluenced by reducing the number of samples to the point where the period between two consecutive samples measured in Courant–Friedrichss–Levy (CFL) condition units is below one. Nevertheless, crossing this limit, the estimates of uncertainties start to exhibit significant growth.

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