Numerical Investigation of Conformal ADI-FDTD Schemes with Second-Order Convergence

2010 ◽  
Vol E93-C (1) ◽  
pp. 52-59 ◽  
Author(s):  
Kazuhiro FUJITA ◽  
Yoichi KOCHIBE ◽  
Takefumi NAMIKI
Keyword(s):  
Numerical Investigation ◽  
Second Order ◽  
Order Convergence ◽  
Second Order Convergence

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

scholarly journals Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity

Processes ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 215 ◽  
Author(s):  
Zhanjie Song ◽  
Yaxuan Xing ◽  
Qingzhi Hou ◽  
Wenhuan Lu
Keyword(s):  
Heat Conduction ◽  
Convergence Rate ◽  
Smoothed Particle Hydrodynamics ◽  
Transient Heat Conduction ◽  
Second Order ◽  
Order Convergence ◽  
Particle Hydrodynamics ◽  
Smoothed Particle ◽  
Initial Discontinuity ◽  
Second Order Convergence

To eliminate the numerical oscillations appearing in the first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method for simulating transient heat conduction problems with discontinuous initial distribution, this paper presents a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method. Numerical properties of both SO-SSPH and FO-SSPH are analyzed, including truncation error, numerical accuracy, convergence rate, and stability. Experimental results show that for transient heat conduction with initial smooth distribution, both FO-SSPH and SO-SSPH can achieve second order convergence rate, which is consistent with the theoretical analysis. However, for one- and two-dimensional conduction with initial discontinuity, the FO-SSPH method suffers from serious unphysical oscillations, which do not disappear over time, and hence it only achieves a first-order convergence rate; while the present SO-SSPH method can avoid unphysical oscillations and has second-order convergence rate. Therefore, the SO-SSPH method is a feasible tool for solving transient heat conduction problems with both smooth and discontinuous distributions, and it is easy to be extended to high dimensional cases.

Sign in / Sign up

Export Citation Format

Share Document