High Order Accuracy Computational Methods for Long Time Integration of Nonlinear PDEs in Complex Domains

1998 ◽  
Author(s):  
David Gottlieb ◽  
C. W. Shu ◽  
P. F. Fischer ◽  
W. S. Don ◽  
J. Hesthaven
Keyword(s):  
Computational Methods ◽  
Time Integration ◽  
High Order ◽  
Nonlinear Pdes ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Long Time ◽  
Long Time Integration

scholarly journals A flexible symplectic scheme for two-dimensional Schrödinger equation with highly accurate RBFS quasi-interpolation

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5451-5461 ◽  
Author(s):  
Shengliang Zhang ◽  
Liping Zhang
Keyword(s):  
Numerical Experiments ◽  
Time Integration ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Order Accuracy ◽  
Symplectic Scheme ◽  
Long Time ◽  
Long Time Integration

Based on highly accurate multiquadric quasi-interpolation, this study suggests a meshless symplectic procedure for two-dimensional time-dependent Schr?dinger equation. The method is highorder accurate, flexible with respect to the geometry, computationally efficient and easy to implement. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good performance in the long time integration.

scholarly journals Symplectic multiquadric quasi-interpolation approximations of KdV equation

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5161-5171
Author(s):  
Shengliang Zhang ◽  
Liping Zhang
Keyword(s):  
Kdv Equation ◽  
Order Approximation ◽  
High Order ◽  
Basis Functions ◽  
High Order Accuracy ◽  
Approximation Properties ◽  
High Order Approximation ◽  
Shape Preserving ◽  
Order Accuracy ◽  
Long Time

Radial basis functions quasi-interpolation is very useful tool for the numerical solution of differential equations, since it possesses shape-preserving and high-order approximation properties. Based on multiquadric quasi-interpolations, this study suggests a meshless symplectic procedure for KdV equation. The method has a number of advantages over existing approaches including no need to solve a resultant full matrix, accuracy and ease of implementation. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good property of long-time tracking capability.

scholarly journals Clutter Cancellation and Long Time Integration for GNSS-Based Passive Bistatic Radar

2021 ◽  
Vol 13 (4) ◽  
pp. 701 ◽  
Author(s):  
Binbin Wang ◽  
Hao Cha ◽  
Zibo Zhou ◽  
Bin Tian
Keyword(s):  
Fourier Transform ◽  
Target Detection ◽  
Time Integration ◽  
Detection Performance ◽  
Bistatic Radar ◽  
Detection Range ◽  
Robust Detection ◽  
Coherent Integration ◽  
Long Time ◽  
Long Time Integration

Clutter cancellation and long time integration are two vital steps for global navigation satellite system (GNSS)-based bistatic radar target detection. The former eliminates the influence of direct and multipath signals on the target detection performance, and the latter improves the radar detection range. In this paper, the extensive cancellation algorithm (ECA), which projects the surveillance channel signal in the subspace orthogonal to the clutter subspace, is first applied in GNSS-based bistatic radar. As a result, the clutter has been removed from the surveillance channel effectively. For long time integration, a modified version of the Fourier transform (FT), called long-time integration Fourier transform (LIFT), is proposed to obtain a high coherent processing gain. Relative acceleration (RA) is defined to describe the Doppler variation results from the motion of the target and long integration time. With the estimated RA, the Doppler frequency shift compensation is carried out in the LIFT. This method achieves a better and robust detection performance when comparing with the traditional coherent integration method. The simulation results demonstrate the effectiveness and advantages of the proposed processing method.

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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