scholarly journals L 1 -Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jian Chen ◽  
Yong Huang ◽  
Taishan Zeng
Keyword(s):  
Burgers Equation ◽  
Computational Cost ◽  
Polynomial Space ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Discrete Scheme ◽  
Optimal Convergence ◽  
Fully Discrete ◽  
Convergence Orders

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by L 1 -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach O τ 2 − α + h r in the L 2 norm and O τ 2 − α + h r − 1 in the H 1 norm, respectively, in which τ is the time step size, h is the space step size, and r is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.

scholarly journals Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

2021 ◽  
Vol 41 (1) ◽  
Author(s):  
Higinio Ramos ◽  
Anurag Kaur ◽  
V. Kanwar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Burgers Equation ◽  
Block Method ◽  
Step Size ◽  
Time Step ◽  
Partial Differential ◽  
Time Step Size ◽  
Linear Partial Differential Equations ◽  
Non Linear

AbstractIn this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.

scholarly journals Brief communication: Time step dependence (and fixes) in Stokes simulations of calving ice shelves

2020 ◽  
Vol 14 (9) ◽  
pp. 3209-3213
Author(s):  
Brandon Berg ◽  
Jeremy Bassis
Keyword(s):  
Boundary Condition ◽  
Computational Cost ◽  
Hydrostatic Equilibrium ◽  
Stress Fields ◽  
Step Size ◽  
Time Step ◽  
Ice Shelves ◽  
Time Step Size ◽  
Communication Time ◽  
Rapid Changes

Abstract. The buoyancy boundary condition applied to floating portions of ice sheets and glaciers in Stokes models requires special consideration when the glacier rapidly departs from hydrostatic equilibrium. This boundary condition can manifest in velocity fields that are unphysically (and strongly) dependent on time step size, thereby contaminating diagnostic stress fields. This can be especially problematic for models of calving glaciers, where rapid changes in geometry cause configurations that suddenly depart from hydrostatic equilibrium and lead to inaccurate estimates of the stress field. Here we show that the unphysical behavior can be cured with minimal computational cost by reintroducing a regularization that corresponds to the acceleration term in the stress balance. This regularization provides consistent velocity solutions for all time step sizes.

scholarly journals Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

2021 ◽  
Author(s):  
Charles-Edouard Bréhier
Keyword(s):  
Time Horizon ◽  
Polynomial Growth ◽  
Computational Cost ◽  
Invariant Distribution ◽  
Euler Scheme ◽  
Time Behavior ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Moment Bounds

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

scholarly journals Brief communication: Time step dependence (and fixes) in Stokes simulations of calving ice shelves

2020 ◽  
Author(s):  
Brandon Berg ◽  
Jeremy Bassis
Keyword(s):  
Computational Cost ◽  
Hydrostatic Equilibrium ◽  
Stress Fields ◽  
Step Size ◽  
Time Step ◽  
Ice Shelves ◽  
Time Step Size ◽  
Communication Time ◽  
Rapid Changes ◽  
Ill Posed

Abstract. The buoyancy boundary condition applied to floating portions of ice sheets and glaciers in Stokes models is numerically ill-posed when the glacier rapidly departs from hydrostatic equilibrium. This manifests in velocity solutions that diverge with decreasing time step size, contaminating diagnostic strain rate and stress fields. This can be especially problematic for models of calving glaciers, where rapid changes in geometry lead to configurations that depart from hydrostatic equilibrium and accurate measures of the stress field are needed. Here we show that the singular behavior can be cured with minimal computational cost by reintroducing a regularization that corresponds to the acceleration term in the stress balance. This regularization provides numerically stable velocity solutions for all time step sizes.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

2021 ◽  
Author(s):  
Ethan Corle ◽  
Matthew Floros ◽  
Sven Schmitz
Keyword(s):  
Experimental Data ◽  
Particle Method ◽  
Comprehensive Analysis ◽  
Solution Stability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Vortex Particle Method ◽  
Whirl Flutter ◽  
Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Numerical Simulation of Proppant Transport in Hydraulically Fractured Reservoirs

2021 ◽  
Author(s):  
Seyhan Emre Gorucu ◽  
Vijay Shrivastava ◽  
Long X. Nghiem
Keyword(s):  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Fractured Reservoirs ◽  
Injection Timing ◽  
Fracture Permeability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Proppant Transport ◽  
Fracture Closure

Abstract An existing equation-of-state compositional simulator is extended to include proppant transport. The simulator determines the final location of the proppant after fracture closure, which allows the computation of the permeability along the hydraulic fracture. The simulation then continues until the end of the production. During hydraulic fracturing, proppant is injected in the reservoir along with water and additives like polymers. Hydraulic fracture gets created due to change in stress caused by the high injection pressure. Once the fracture opens, the bulk slurry moves along the hydraulic fracture. Proppant moves at a different speed than the bulk slurry and sinks down by gravity. While the proppant flows along the fracture, some of the slurry leaks off into the matrix. As the fracture closes after injection stops, the proppant becomes immobile. The immobilized proppant prevents the fracture from closing and thus keeps the permeability of the fracture high. All the above phenomena are modelled effectively in this new implementation. Coupled geomechanics simulation is used to model opening and closure of the fracture following geomechanics criteria. Proppant retardation, gravitational settling and fluid leak-off are modeled with the appropriate equations. The propped fracture permeability is a function of the concentration of immobilized proppant. The developed proppant simulation feature is computationally stable and efficient. The time step size during the settling adapts to the settling velocity of the proppants. It is found that the final location of the proppants is highly dependent on its volumetric concentration and slurry viscosity due to retardation and settling effects. As the location and the concentration of the proppants determine the final fracture permeability, the additional feature is expected to correctly identify the stimulated region. In this paper, the theory and the model formulation are presented along with a few key examples. The simulation can be used to design and optimize the amount of proppant and additives, injection timing, pressure, and well parameters required for successful hydraulic fracturing.

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