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Yan Wang - One of the best experts on this subject based on the ideXlab platform.
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a uniformly accurate ua multiscale time integrator pseudospectral method for the nonlinear dirac equation in the Nonrelativistic Limit regime
Mathematical Modelling and Numerical Analysis, 2018Co-Authors: Yongyong Cai, Yan WangAbstract:A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter e ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (e 2 ) and O (1) in time and space, respectively. In the Nonrelativistic regime, i.e. , 0 ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds in e ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as and h m 0 + τ 2 + e 2 , where h is the mesh size, τ is the time step and m 0 depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ ) for all e ∈ (0, 1] and optimally with quadratic convergence rate at O (τ 2 ) in the regimes when either e = O (1) or 0 ≲ τ . Numerical results are reported to demonstrate that our error estimates are optimal and sharp.
Weizhu Bao - One of the best experts on this subject based on the ideXlab platform.
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numerical methods and comparison for the dirac equation in the Nonrelativistic Limit regime
Journal of Scientific Computing, 2017Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Qinglin TangAbstract:We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the Nonrelativistic Limit regime, involving a small dimensionless parameter $$0<\varepsilon \ll 1$$0<źź1 which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $$O(\varepsilon ^2)$$O(ź2) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the Nonrelativistic Limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step $$\tau $$ź as well as the small parameter $$\varepsilon $$ź. Based on the error bounds, in order to obtain `correct' numerical solutions in the Nonrelativistic Limit regime, i.e. $$0<\varepsilon \ll 1$$0<źź1, the FDTD methods share the same $$\varepsilon $$ź-scalability on time step and mesh size as: $$\tau =O(\varepsilon ^3)$$ź=O(ź3) and $$h=O(\sqrt{\varepsilon })$$h=O(ź). Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the symmetric exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $$\varepsilon $$ź-scalability is improved to $$\tau =O(\varepsilon ^2)$$ź=O(ź2) and $$h=O(1)$$h=O(1) when $$0<\varepsilon \ll 1$$0<źź1. Extensive numerical results are reported to support our error estimates.
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a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the Nonrelativistic Limit regime
Numerische Mathematik, 2017Co-Authors: Weizhu Bao, Xiaofei ZhaoAbstract:A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein---Gordon---Schrodinger (KGS) equations in the Nonrelativistic Limit regime with a dimensionless parameter $$0<\varepsilon \le 1$$0<ź≤1 which is inversely proportional to the speed of light. In fact, the solution of the KGS equations propagates waves with wavelength at $$O(\varepsilon ^2)$$O(ź2) and O(1) in time and space, respectively, when $$0<\varepsilon \ll 1$$0<źź1, which brings significantly numerical burden in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at $$O(\tau ^2/\varepsilon ^2+h^{m_0})$$O(ź2/ź2+hm0) and $$O(\varepsilon ^2+h^{m_0})$$O(ź2+hm0) for $$\varepsilon \in (0,1]$$źź(0,1] with $$\tau $$ź time step size, h mesh size and $$m_0\ge 4$$m0ź4 an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $$O(\tau )$$O(ź) for $$\varepsilon \in (0,1]$$źź(0,1]. In addition, the MTI-FP method converges optimally with quadratic convergence rate at $$O(\tau ^2)$$O(ź2) in the regime when $$0<\tau \lesssim \varepsilon ^2$$0<źźź2 and the error is at $$O(\varepsilon ^2)$$O(ź2) independent of $$\tau $$ź in the regime when $$0<\varepsilon \lesssim \tau ^{1/2}$$0<źźź1/2. Thus the meshing strategy requirement (or $$\varepsilon $$ź-scalability) of the MTI-FP is $$\tau =O(1)$$ź=O(1) and $$h=O(1)$$h=O(1) for $$0<\varepsilon \ll 1$$0<źź1, which is significantly better than that of classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to its Limiting models in the Nonrelativistic Limit regime.
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error estimates of numerical methods for the nonlinear dirac equation in the Nonrelativistic Limit regime
Science China-mathematics, 2016Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Jia YinAbstract:We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation (NLDE) in the Nonrelativistic Limit regime, involving a small dimensionless parameter 0 < e ≤ 1 which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(e 2) and O(1) in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 < e ≤ 1. Based on the error bound, in order to obtain ‘correct’ numerical solutions in the Nonrelativistic Limit regime, i.e., 0 < e ≤ 1, the CNFD method requests the e-scalability: τ = O(e 3) and \(h = O\left( {\sqrt \varepsilon } \right)\). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their e-scalability is improved to τ = O(e 2) and h = O(1) when 0 < e ≤ 1. Extensive numerical results are reported to confirm our error estimates.
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a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the Nonrelativistic Limit regime
SIAM Journal on Numerical Analysis, 2016Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Qinglin TangAbstract:We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the Nonrelativistic Limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its Limiting models when $\varepsilon\to0^+$.
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error estimates of numerical methods for the nonlinear dirac equation in the Nonrelativistic Limit regime
arXiv: Numerical Analysis, 2015Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Jia YinAbstract:We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the Nonrelativistic Limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the Nonrelativistic Limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.
Yongyong Cai - One of the best experts on this subject based on the ideXlab platform.
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a uniformly accurate ua multiscale time integrator pseudospectral method for the nonlinear dirac equation in the Nonrelativistic Limit regime
Mathematical Modelling and Numerical Analysis, 2018Co-Authors: Yongyong Cai, Yan WangAbstract:A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter e ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (e 2 ) and O (1) in time and space, respectively. In the Nonrelativistic regime, i.e. , 0 ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds in e ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as and h m 0 + τ 2 + e 2 , where h is the mesh size, τ is the time step and m 0 depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ ) for all e ∈ (0, 1] and optimally with quadratic convergence rate at O (τ 2 ) in the regimes when either e = O (1) or 0 ≲ τ . Numerical results are reported to demonstrate that our error estimates are optimal and sharp.
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numerical methods and comparison for the dirac equation in the Nonrelativistic Limit regime
Journal of Scientific Computing, 2017Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Qinglin TangAbstract:We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the Nonrelativistic Limit regime, involving a small dimensionless parameter $$0<\varepsilon \ll 1$$0<źź1 which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $$O(\varepsilon ^2)$$O(ź2) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the Nonrelativistic Limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step $$\tau $$ź as well as the small parameter $$\varepsilon $$ź. Based on the error bounds, in order to obtain `correct' numerical solutions in the Nonrelativistic Limit regime, i.e. $$0<\varepsilon \ll 1$$0<źź1, the FDTD methods share the same $$\varepsilon $$ź-scalability on time step and mesh size as: $$\tau =O(\varepsilon ^3)$$ź=O(ź3) and $$h=O(\sqrt{\varepsilon })$$h=O(ź). Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the symmetric exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $$\varepsilon $$ź-scalability is improved to $$\tau =O(\varepsilon ^2)$$ź=O(ź2) and $$h=O(1)$$h=O(1) when $$0<\varepsilon \ll 1$$0<źź1. Extensive numerical results are reported to support our error estimates.
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error estimates of numerical methods for the nonlinear dirac equation in the Nonrelativistic Limit regime
Science China-mathematics, 2016Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Jia YinAbstract:We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation (NLDE) in the Nonrelativistic Limit regime, involving a small dimensionless parameter 0 < e ≤ 1 which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(e 2) and O(1) in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 < e ≤ 1. Based on the error bound, in order to obtain ‘correct’ numerical solutions in the Nonrelativistic Limit regime, i.e., 0 < e ≤ 1, the CNFD method requests the e-scalability: τ = O(e 3) and \(h = O\left( {\sqrt \varepsilon } \right)\). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their e-scalability is improved to τ = O(e 2) and h = O(1) when 0 < e ≤ 1. Extensive numerical results are reported to confirm our error estimates.
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a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the Nonrelativistic Limit regime
SIAM Journal on Numerical Analysis, 2016Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Qinglin TangAbstract:We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the Nonrelativistic Limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its Limiting models when $\varepsilon\to0^+$.
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error estimates of numerical methods for the nonlinear dirac equation in the Nonrelativistic Limit regime
arXiv: Numerical Analysis, 2015Co-Authors: Weizhu Bao, Yongyong Cai, Xiaowei Jia, Jia YinAbstract:We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the Nonrelativistic Limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this Limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the Nonrelativistic Limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.
Horatiu Nastase - One of the best experts on this subject based on the ideXlab platform.
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Nonrelativistic Limit of the abelianized abjm model and the ads cmt correspondence
Journal of High Energy Physics, 2016Co-Authors: Cristhiam Lopezarcos, Jeff Murugan, Horatiu NastaseAbstract:We consider the Nonrelativistic Limit of the abelian reduction of the massive ABJM model proposed in [1], obtaining a supersymmetric version of the Jackiw-Pi model. The system exhibits an $$ \mathcal{N}=2 $$ Super-Schrodinger symmetry with the Jackiw-Pi vortices emerging as BPS solutions. We find that this (2 + 1)-dimensional abelian field theory is dual to a certain (3+1)-dimensional gravity theory that differs somewhat from previously considered abelian condensed matter stand-ins for the ABJM model. We close by commenting on progress in the top-down realization of the AdS/CMT correspondence in a critical string theory.
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Nonrelativistic Limit of the abelianized abjm model and the ads cmt correspondence
Journal of High Energy Physics, 2016Co-Authors: Cristhiam Lopezarcos, Jeff Murugan, Horatiu NastaseAbstract:The Laboratory for Quantum Gravity & Strings Department of Mathematics and Applied Mathematics University of Cape Town, Private Bag
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Nonrelativistic Limit of the abelianized abjm model and the ads cmt correspondence
arXiv: High Energy Physics - Theory, 2015Co-Authors: Cristhiam Lopezarcos, Jeff Murugan, Horatiu NastaseAbstract:We consider the Nonrelativistic Limit of the abelian reduction of the massive ABJM model proposed in \cite{Mohammed:2012gi}, obtaining a supersymmetric version of the Jackiw-Pi model. The system exhibits an ${\cal N}=2$ Super-Schr\"odinger symmetry with the Jackiw-Pi vortices emerging as BPS solutions. We find that this $(2+1)$-dimensional abelian field theory is dual to a certain (3+1)-dimensional gravity theory that differs somewhat from previously considered abelian condensed matter stand-ins for the ABJM model. We close by commenting on progress in the top-down realization of the AdS/CMT correspondence in a critical string theory.
Xiaofei Zhao - One of the best experts on this subject based on the ideXlab platform.
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a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the Nonrelativistic Limit regime
Numerische Mathematik, 2017Co-Authors: Weizhu Bao, Xiaofei ZhaoAbstract:A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein---Gordon---Schrodinger (KGS) equations in the Nonrelativistic Limit regime with a dimensionless parameter $$0<\varepsilon \le 1$$0<ź≤1 which is inversely proportional to the speed of light. In fact, the solution of the KGS equations propagates waves with wavelength at $$O(\varepsilon ^2)$$O(ź2) and O(1) in time and space, respectively, when $$0<\varepsilon \ll 1$$0<źź1, which brings significantly numerical burden in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at $$O(\tau ^2/\varepsilon ^2+h^{m_0})$$O(ź2/ź2+hm0) and $$O(\varepsilon ^2+h^{m_0})$$O(ź2+hm0) for $$\varepsilon \in (0,1]$$źź(0,1] with $$\tau $$ź time step size, h mesh size and $$m_0\ge 4$$m0ź4 an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $$O(\tau )$$O(ź) for $$\varepsilon \in (0,1]$$źź(0,1]. In addition, the MTI-FP method converges optimally with quadratic convergence rate at $$O(\tau ^2)$$O(ź2) in the regime when $$0<\tau \lesssim \varepsilon ^2$$0<źźź2 and the error is at $$O(\varepsilon ^2)$$O(ź2) independent of $$\tau $$ź in the regime when $$0<\varepsilon \lesssim \tau ^{1/2}$$0<źźź1/2. Thus the meshing strategy requirement (or $$\varepsilon $$ź-scalability) of the MTI-FP is $$\tau =O(1)$$ź=O(1) and $$h=O(1)$$h=O(1) for $$0<\varepsilon \ll 1$$0<źź1, which is significantly better than that of classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to its Limiting models in the Nonrelativistic Limit regime.
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a uniformly accurate multiscale time integrator pseudospectral method for the klein gordon equation in the Nonrelativistic Limit regime
arXiv: Numerical Analysis, 2015Co-Authors: Weizhu Bao, Yongyong Cai, Xiaofei ZhaoAbstract:We propose and analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for solving the Klein-Gordon (KG) equation with a dimensionless parameter $0<\varepsilon\leq1$ which is inversely proportional to the speed of light. In the Nonrelativistic Limit regime, i.e. $0<\varepsilon\ll1$, the solution to the KG equation propagates waves with amplitude at $O(1)$ and wavelength at $O(\varepsilon^2)$ in time and $O(1)$ in space, which causes significantly numerical burdens due to the high oscillation in time. The MTI-FP method is designed by adapting a multiscale decomposition by frequency (MDF) to the solution at each time step and applying an exponential wave integrator to the nonlinear Schrodinger equation with wave operator under well-prepared initial data for $\varepsilon^2$-frequency and $O(1)$-amplitude waves and a KG-type equation with small initial data for the reminder waves in the MDF. We rigorously establish two independent error bounds in $H^2$-norm to the MTI-FP method at $O(h^{m_0}+\tau^2+\varepsilon^2)$ and $O(h^{m_0}+\tau^2/\varepsilon^2)$ with $h$ mesh size, $\tau$ time step and $m_0\ge2$ an integer depending on the regularity of the solution, which immediately imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\le \tau$. Numerical results are reported to confirm the error bounds and demonstrate the efficiency and accuracy of the MTI-FP method for the KG equation, especially in the Nonrelativistic Limit regime.
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a uniformly accurate ua multiscale time integrator fourier pseoduspectral method for the klein gordon schrodinger equations in the Nonrelativistic Limit regime
arXiv: Numerical Analysis, 2015Co-Authors: Weizhu Bao, Xiaofei ZhaoAbstract:A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein-Gordon-Schr\"{o}dinger (KGS) equations in the Nonrelativistic Limit regime with a dimensionless parameter $0<\varepsilon\le1$ which is inversely proportional to the speed of light. In fact, the solution to the KGS equations propagates waves with wavelength at $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively, when $0<\varepsilon\ll 1$, which brings significantly numerical burdens in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency to the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at $O(\tau^2/\varepsilon^2+h^{m_0})$ and $O(\varepsilon^2+h^{m_0})$ for $\varepsilon\in(0,1]$ with $\tau$ time step size, $h$ mesh size and $m_0\ge 4$ an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regime when either $\varepsilon=O(1)$ or $0<\varepsilon\le \tau$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) of the MTI-FP is $\tau=O(1)$ and $h=O(1)$ for $0<\varepsilon\ll 1$, which is significantly better than classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to the Limiting models in the Nonrelativistic Limit regime.
