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1.
Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations 下载免费PDF全文
In this paper, two multiscale time integrators (MTIs), motivated from two
types of multiscale decomposition by either frequency or frequency and amplitude, are
proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0 < \varepsilon≤ 1.$ In fact, the solution to this equation
propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon≪1,$ which brings significantly
numerical burdens in practical computation. We rigorously establish two independent
error bounds for the two MTIs at $O(\tau^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon ∈ (0,1]$ with $\tau > 0$ as step
size, which imply that the two MTIs converge uniformly with linear convergence rate
at $O(\tau)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the
regimes when either $ε=O(1)$ or $0<ε≤\tau.$ Thus the meshing strategy requirement (or $ε$-scalability) of the two MTIs is $\tau =O(1)$ for $0<ε≪1,$ which is significantly improved
from $\tau =O(ε^3)$ and $\tau =O(ε^2)$ requested by finite difference methods and exponential
wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards
better understanding on the convergence and resolution properties of the two MTIs.
In addition, numerical results support the two error bounds very well. 相似文献
2.
Zhaowei Tian Shuying Zhai Zhifeng Weng 《Journal of Applied Analysis & Computation》2020,10(3):904-919
In this paper, three compact difference schemes for the time-fractional Black-Scholes model governing European option pricing are presented. Firstly, in order to obtain the fourth-order accuracy in space by applying the Pad\''{e} approximation, we eliminate the convection term of the B-S equation by an exponential transformation. Then the time fractional derivative is approximated by $L1$ formula, $L2 - 1_\sigma$ formula and $L1 - 2$ formula respectively, and three compact difference schemes with oders $O(\Delta t^{2-\alpha}+h ^4)$, $O(\Delta t^{2}+h ^4)$ and $O(\Delta t^{3-\alpha}+h ^4)$ are constructed. Finally, numerical example is carried out to verify the accuracy and effectiveness of proposed methods, and the comparisons of various schemes are given. The paper also provides numerical studies including the effect of fractional orders and the effect of different parameters on option price in time-fractional B-S model. 相似文献
3.
Sheng Zhang & Hong-Ci Huang 《计算数学(英文版)》1991,9(1):17-27
The domain decomposition method in this paper is based on PCG (Preconditioned Conjugate Gradient method). If $N$ is the number of subdomains, the number of sub-problems solved parallelly in a PCG step is $\frac{4}{3}(1-\frac{1}{4^{\log N+1}})N$. The condition number of the preconditioned system does not exceed $O(1+\log N)^3$. It is completely independent of the mesh size. The number of iterations required, to decrease the energy norm of the error by a fixed factor, is proportional to $O(1+\log N)^{\frac{3}{2}}$ . 相似文献
4.
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 ε≤ 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(ε~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 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). 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 timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates. 相似文献
5.
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end , with a finite unknown ‘blow‐up’ time Tb have been studied in a previous work. Specifically, for ε a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ‘mass control’ property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve‐\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon‐u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end , \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow‐up time Tb and the blow‐up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end , with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end . Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end , we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end , defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end . In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end , the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end , verifies, $\|(u‐u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1‐\alpha}$\nopagenumbers\end , \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end . The two special cases of a single‐point blow‐up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p‐1})$\nopagenumbers\end . The estimate $|T_\eta‐T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
6.
In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm. 相似文献
7.
Guo-Rong Wang & Sen-Quan Lu 《计算数学(英文版)》1988,6(4):348-354
The parallel arithmetic complexities for computing generalized inverse $A^+$, computing the minimum-norm least-squares solution of $Ax=b$, computing order $m+n-r$ determinants and finding the characteristic polynomials of order $m+n-r$ matrices are shown to have the same grawth rate. Algorithms are given that compute $A^+$ and $A_{MN}^+$ in $O(\log r\dot \log n+\log m)$ and $O(\log^2n+\log m)$ steps using a number of processors which is a polynomial in $m, \ n$ and $r$ $(A\in B_r^{m\times n},r=rank \ A)$. 相似文献
8.
Sunyoung Bu 《Journal of Applied Analysis & Computation》2016,6(2):443-462
Error correction method (ECM)~\cite{kim2011a,kim2011b} which has been
recently developed, is based on the construction of a local
approximation to the solution on each time step, and has the
excellent convergence order $O(h^{2p+2})$, provided the local
approximation has a local residual error $O(h^p)$.
In this paper, we construct a higher-order continuous local platform
to develop higher-order semi-explicit one-step ECM for solving initial
value time dependent differential equations. It is shown that special
choices of parameters for the local platform can lead to
the improvement of the well-known explicit fourth and fifth order
Runge-Kutta methods. Numerical experiments demonstrate
the theoretical results 相似文献
9.
The interior Dirichlet problem for Laplace's equation on a plane polygonal region $\Omega$ with boundary $\Gamma$ may be reformulated as a second kind integral equation on $\Gamma$. This equation may be solved by the Nyström method using the composite trapezoidal rule. It is known that if the mesh has $O(n)$ points and is graded appropriately, then $O(1/n^2)$ convergence is obtained for the solution of the integral equation and the associated solution to the Dirichlet problem at any $x\in \Omega$. We present a simple extrapolation scheme which increases these rates of convergence to $O(1/n^4)$ . 相似文献
10.
EXPANSIONS OF STEP-TRANSITION OPERATORS OF MULTI-STEP METHODS AND ORDER BARRIERS FOR DAHLQUIST PAIRS
Quan-dong Feng Yi-fa Tang 《计算数学(英文版)》2006,24(1):45-58
Using least parameters, we expand the step-transition operator of any linear multi-step method (LMSM) up to O(τ^s+5) with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 (obtained by the second author in a former paper). We prove that in the conjugate relation G3^λτ o G1^τ =G2^τ o G3^λτ with G1 being an LMSM,(1) theorder of G2 can not be higher than that of G1; (2) if G3 is also an LMSM and G2 is a symplectic B-series, then the orders of G1, G2 and G3 must be 2, 2 and 1 respectively. 相似文献
11.
管克英 《应用数学学报(英文版)》1984,1(1):8-16
In this article,we consider the following nonlinear Sturm-Liouville problem(?)and prove the existence of the eigenvalue and the eigenfunction by using Schauder's fixed opinttheorem.This problem arises from finding the solutions of solitons and stationary states of thenonlinear Schr(?)dinger equation (NLS Eq.) with external fields.Using the result obtained,we provethe existence of solitons and stationary states of the NLS equation for the oscillater. 相似文献
12.
Superconvergence Analysis of a BDF-Galerkin FEM for the Nonlinear Klein-Gordon-Schrödinger Equations with Damping Mechanism 下载免费PDF全文
Dongyang Shi & Houchao Zhang 《计算数学(英文版)》2023,41(2):224-245
The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting'' technique, the superconvergence behavior of order $O(h^2+\tau^2)$ in $H^1$-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, $h$ is the spatial subdivision parameter, and $\tau$ is the time step. 相似文献
13.
Finite element algorithm based on high-order time approximation for time fractional convection-diffusion equation 下载免费PDF全文
Xinfei Liu Yang Liu Hong Li Zhichao Fang Jinfeng Wang 《Journal of Applied Analysis & Computation》2018,8(1):229-249
In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result $O(h^{r+1}+\tau^{3-\alpha})$ can be derived, which illustrates that time convergence rate is higher than the order $(2-\alpha)$ derived by $L1$-approximation. Finally, to validate our theoretical results, some computing data are provided. 相似文献
14.
我们考虑了一类原型为$$\begin{cases}u_t-\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma|\nabla u|^2-\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega,\end{cases}$$的一类抛物方程. 其中, 函数$|\overrightarrow{b}(x,t)|^2,|\overrightarrow{F}(x,t)|^2,f(x,t)$位于空间$L^r{(0,T;L^q(\Omega))}$, $\gamma$是一个正常数. 在源项和梯度的系数项在空间$L^r{(0,T;L^q(\Omega))}$具有合适的可积条件下, 本文的目的在于证明先验的$L^\infty$估计以及方程存在有界解. 主要的方法包括通过正则化建立扰动问题, 用非线性的检验函数实现Stampacchia迭代技术以及极限过程中的紧性论断. 相似文献
15.
16.
Isao Shoji. 《Mathematics of Computation》1998,67(221):287-298
This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be and in the sense respectively, where is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.
17.
18.
研究一类二阶线性脉冲微分方程解的结构和解的渐近性态,其中δ(t)是δ-函数,且对n∈N有an>0,r(t)>0是[t0, ∞) 上的连续函数,0≤t0相似文献
19.
考虑由未知二元函数的近似值计算其Laplace算子与二阶混合偏导数的问题,给出稳定逼近Laplace算子与二阶混合偏导数的两类Lanczos方法,其逼近精度分别为O(δ~(1/2))和O(δ~(2/3)),其中δ是近似函数的误差水平. 相似文献
20.
Markus Keel Hart F. Smith Christopher D. Sogge 《Journal of the American Mathematical Society》2004,17(1):109-153
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only use the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the decay of solutions of the wave equation as much as the decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equation that gives decay of solutions of the inhomogeneous linear wave equation in terms of a -weighted norm on the forcing term. A weighted space-time estimate for inhomogeneous wave equations is also important in making the spatial decay useful for the long-term existence argument.