A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the eval...A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.展开更多
The partial differential equation(PDE)solution of the telegrapher is a promising fault location method among time-domain and model-based techniques.Recent research works have shown that the leap-frog process is superi...The partial differential equation(PDE)solution of the telegrapher is a promising fault location method among time-domain and model-based techniques.Recent research works have shown that the leap-frog process is superior to other explicit methods for the PDE solution.However,its implementation is challenged by determining the initial conditions in time and the boundary conditions in space.This letter proposes two implicit solution methods for determining the initial conditions and an analytical way to obtain the boundary conditions founded on the signal decomposition.The results show that the proposal gives fault location accuracy superior to the existing leap-frog scheme,particularly in the presence of harmonics.展开更多
An economical explicit scheme of time integration is implemented in a regional model over Indian region to achieve computational economy. The model is also integrated by explicit Leap-Frog Scheme. The performance of e...An economical explicit scheme of time integration is implemented in a regional model over Indian region to achieve computational economy. The model is also integrated by explicit Leap-Frog Scheme. The performance of economical explicit scheme is evaluated by comparing the forecast results with those produced by leap-frog scheme. The results show that the economical explicit scheme produces more or less similar forecasts as compared to those produced with leap-frog scheme. However, application of the economical explicit scheme saves substantial amount of computer time. The scheme is found nearly four times economical as compared to explicit leap-frog scheme.展开更多
In this paper,the convergence and stability of the ’Leap-frog’ finite difference scheme for the semilinear wave equation are proved by using of the bounded extensive method under more generalized condition for the n...In this paper,the convergence and stability of the ’Leap-frog’ finite difference scheme for the semilinear wave equation are proved by using of the bounded extensive method under more generalized condition for the nonlinear term. The more complex standard a priori estimates are avoided so that the theoretical results are complemented for the scheme which was presented by Perring and Skyrne (1962).展开更多
In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By...In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.展开更多
In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative ...In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.展开更多
We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artific...We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.展开更多
基金supported by a grant from the French National Ministry of Education and Research(MENSR,19755-2005)
文摘A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.
基金Izudin Dzafic was supported by the Federal Ministry of Education and ScienceBosniathrough funding
文摘The partial differential equation(PDE)solution of the telegrapher is a promising fault location method among time-domain and model-based techniques.Recent research works have shown that the leap-frog process is superior to other explicit methods for the PDE solution.However,its implementation is challenged by determining the initial conditions in time and the boundary conditions in space.This letter proposes two implicit solution methods for determining the initial conditions and an analytical way to obtain the boundary conditions founded on the signal decomposition.The results show that the proposal gives fault location accuracy superior to the existing leap-frog scheme,particularly in the presence of harmonics.
文摘An economical explicit scheme of time integration is implemented in a regional model over Indian region to achieve computational economy. The model is also integrated by explicit Leap-Frog Scheme. The performance of economical explicit scheme is evaluated by comparing the forecast results with those produced by leap-frog scheme. The results show that the economical explicit scheme produces more or less similar forecasts as compared to those produced with leap-frog scheme. However, application of the economical explicit scheme saves substantial amount of computer time. The scheme is found nearly four times economical as compared to explicit leap-frog scheme.
文摘In this paper,the convergence and stability of the ’Leap-frog’ finite difference scheme for the semilinear wave equation are proved by using of the bounded extensive method under more generalized condition for the nonlinear term. The more complex standard a priori estimates are avoided so that the theoretical results are complemented for the scheme which was presented by Perring and Skyrne (1962).
基金supported by the NSF of China(Grant Nos.11801527,11701522,11771163,12011530058,11671160,1191101330)by the China Postdoctoral Science Foundation(Grant Nos.2018M632791,2019M662506).
文摘In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.
文摘In this paper, the periodic initial value problem for the following class of nonlinear Schrodinger equation of high order i partial derivative u/partial derivative t + (-1)(m) partial derivative(m)/partial derivative x(m) (a(x)partial derivative(m)u/partial derivative x(m)) + beta(x)q(\u\(2))u + f(x, t)u = g(x, t) is considered. A leap-frog finite difference scheme is given, and convergence and stability is proved. Finally, it is shown by a numerical example that numerical result is coincident with theoretical result.
基金Singapore A*STAR SERC PSF-Grant No.1321202067National Natural Science Foundation of China Grant NSFC41390452the Doctoral Programme Foundation of Institution of Higher Education of China as well as by the Austrian Science Foundation(FWF)under grant No.F41(project VICOM)and grant No.I830(project LODIQUAS)and grant No.W1245 and the Austrian Ministry of Science and Research via its grant for the WPI.
文摘We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.
