An enhanced FIVER method for multi-material flow problems with second-order convergence rate

2017 ◽  
Vol 329 ◽  
pp. 141-172 ◽  
Author(s):  
Alex Main ◽  
Xianyi Zeng ◽  
Philip Avery ◽  
Charbel Farhat
Keyword(s):  
Convergence Rate ◽  
Material Flow ◽  
Second Order ◽  
Order Convergence ◽  
Flow Problems ◽  
Second Order Convergence

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.

scholarly journals A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 224
Author(s):  
Yang Li ◽  
Yaolei Wang ◽  
Taitao Feng ◽  
Yifei Xin
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Second Order ◽  
Trapezoidal Rule ◽  
Order Convergence ◽  
Integration By Parts Formula ◽  
Order Scheme ◽  
Second Order Convergence

In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.

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.

A lattice Boltzmann model for the nonlinear thermistor equations

2020 ◽  
Vol 31 (03) ◽  
pp. 2050043
Author(s):  
Jiao Liu ◽  
Zhenhua Chai ◽  
Baochang Shi
Keyword(s):  
Convergence Rate ◽  
Lattice Boltzmann ◽  
Nonlinear Diffusion ◽  
Numerical Results ◽  
Lattice Boltzmann Model ◽  
Order Convergence ◽  
Poisson Equations ◽  
Second Order Convergence ◽  
Boltzmann Model ◽  
Good Agreement

In this paper, we propose a general and efficient lattice Boltzmann (LB) model for solving the nonlinear thermistor equations, where the nonlinear diffusion and Poisson equations are solved by two LB equations. Through Chapman–Enskog analysis, the nonlinear thermistor equations can be recovered correctly from the present LB model. We then test the model through some numerical simulations, and find that the numerical results are in good agreement with analytical solutions. Additionally, the numerical results also show that the present LB model has a second-order convergence rate in space.

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