Discontinuous Least-Squares Spatial Discretization Schemes for the One-Dimensional Slab-GeometrySnEquations

2010 ◽  
Vol 164 (3) ◽  
pp. 205-220
Author(s):  
Lei Zhu ◽  
Jim E. Morel
Keyword(s):  
Least Squares ◽  
Spatial Discretization ◽  
One Dimensional ◽  
Discretization Schemes ◽  
The One

Implicit integration factor method for the nonlinear Dirac equation

2018 ◽  
Vol 09 (02) ◽  
pp. 1850019 ◽  
Author(s):  
Jing-Jing Zhang ◽  
Xiang-Gui Li ◽  
Jing-Fang Shao
Keyword(s):  
Numerical Experiments ◽  
Time Discretization ◽  
High Order Accuracy ◽  
Spatial Discretization ◽  
Implicit Integration ◽  
Order Accuracy ◽  
One Dimensional ◽  
Interaction Dynamics ◽  
Nonlinear Dirac Equation ◽  
The One

A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac (NLD) equation. Based on the implicit integration factor (IIF) method, two schemes are proposed. Central differences are applied to the spatial discretization. The semi-discrete scheme keeps the conservation of the charge and energy. For the temporal discretization, second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization. Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves.

scholarly journals Toward error estimates for general space-time discretizations of the advection equation

2020 ◽  
Vol 23 (1-4) ◽  
Author(s):  
Martin J. Gander ◽  
Thibaut Lunet
Keyword(s):  
Error Estimates ◽  
Time Discretization ◽  
Space Time ◽  
Error Tolerance ◽  
Advection Equation ◽  
Order Of Accuracy ◽  
One Dimensional ◽  
General Space ◽  
Discretization Schemes ◽  
The One

AbstractWe develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.

Robust iterative inversion for the one‐dimensional acoustic wave equation

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 357-368 ◽  
Author(s):  
Adam Gersztenkorn ◽  
J. Bee Bednar ◽  
Larry R. Lines
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Robust Regression ◽  
Integral Relation ◽  
Seismic Inversion ◽  
One Dimensional ◽  
Least Squares Solution ◽  
Amplitude Noise ◽  
Lp Norm ◽  
The One

Seismic inversion can be formulated by considering a linearized integral relation which is deduced from the wave equation. This Born inversion approach is equivalent to linear least‐squares inversion for a particular parameterization of the medium. The least‐squares solution is a member of a family of generalized LP norm solutions which are deduced from a maximum‐likelihood formulation. This formulation allows design of various statistical inversion solutions. We present two iterative solutions to the one‐dimensional (1-D) seismic inverse problem: the iterative least‐squares (ILS) and the iterative reweighted least‐squares (IRLS) methods. The ILS method involves solving a distorted background velocity problem after the initial least‐squares solution is obtained. The IRLS method is used as a robust regression technique which is better suited for dealing with certain types of noise and is computationally faster than ILS. Several numerical examples demonstrate that the IRLS method accurately estimates impedance profiles despite the presence of large‐amplitude noise spikes in the seismic traces. Numerical experiments suggest that the IRLS inversion can also be insensitive to noise bursts which are of a lower frequency band than noise spikes.

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