Explicit Large Time Stepping with A Second-Order Exponential Time Integrator Scheme for Unsteady and Steady Flows

2017 ◽  
Author(s):  
Shu-Jie Li ◽  
Zhi J. Wang ◽  
Lili Ju ◽  
Li-Shi Luo
Keyword(s):  
Large Time ◽  
Second Order ◽  
Steady Flows ◽  
Exponential Time ◽  
Time Stepping

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

2012 ◽  
Vol 11 (4) ◽  
pp. 1261-1278 ◽  
Author(s):  
Zhengru Zhang ◽  
Zhonghua Qiao
Keyword(s):  
Difference Scheme ◽  
Numerical Simulations ◽  
Large Time ◽  
Second Order ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Adaptive Time ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

scholarly journals Implicit and semi-implicit second-order time stepping methods for the Richards equation

2021 ◽  
Vol 148 ◽  
pp. 103841
Author(s):  
Sana Keita ◽  
Abdelaziz Beljadid ◽  
Yves Bourgault
Keyword(s):  
Second Order ◽  
Richards Equation ◽  
Time Stepping

scholarly journals General relativistic hydrodynamics on a moving-mesh I: static space–times

2020 ◽  
Vol 496 (1) ◽  
pp. 206-214
Author(s):  
Philip Chang ◽  
Zachariah B Etienne
Keyword(s):  
Initial Pressure ◽  
Second Order ◽  
Moving Mesh ◽  
Central Density ◽  
Relativistic Hydrodynamics ◽  
Time Stepping ◽  
Spatial Reconstruction ◽  
General Relativistic ◽  
Mesh Grid ◽  
Static Space

ABSTRACT We present the moving-mesh general relativistic hydrodynamics solver for static space–times as implemented in the code, MANGA. Our implementation builds on the architectures of MANGA and the numerical relativity python package NRPy+. We review the general algorithm to solve these equations and, in particular, detail the time-stepping; Riemann solution across moving faces; conversion between primitive and conservative variables; validation and correction of hydrodynamic variables; and mapping of the metric to a Voronoi moving-mesh grid. We present test results for the numerical integration of an unmagnetized Tolman–Oppenheimer–Volkoff star for 24 dynamical times. We demonstrate that at a resolution of 106 mesh generating points, the star is stable and its central density drifts downwards by 2 per cent over this time-scale. At a lower resolution, the central density drift increases in a manner consistent with the adopted second-order spatial reconstruction scheme. These results agree well with the exact solutions, and we find the error behaviour to be similar to Eulerian codes with second-order spatial reconstruction. We also demonstrate that the new code recovers the fundamental mode frequency for the same TOV star but with its initial pressure depleted by 10 per cent.

On large time-stepping methods for the Cahn–Hilliard equation

2007 ◽  
Vol 57 (5-7) ◽  
pp. 616-628 ◽  
Author(s):  
Yinnian He ◽  
Yunxian Liu ◽  
Tao Tang
Keyword(s):  
Large Time ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation
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