A simultaneous space-time wavelet method for nonlinear initial boundary value problems

Zhou, Youhe; Wang, Jizeng; Zhang, Lei

doi:10.1007/s10483-018-2384-6

Cited by 9 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Following our previous works [24,25,[31][32][33][34], a continuous function f (x) defined on the interval [0, T] can be approximated by the wavelet expansion…”

Section: Wavelet Approximation Of Multiple Integrals In a Bounded Domainmentioning

confidence: 99%

“…The scaling function ϕ(x) and its derivatives ϕ (i) (x) used in Equation ( 6) can be obtained by using the algorithm given in our previous works [32,34].…”

Section: Wavelet Approximation Of Multiple Integrals In a Bounded Domainmentioning

confidence: 99%

“…Table 1. Coefficients for the wavelet linear multi-step method [32]. Applying Equation ( 21) recursively and using the initial condition directly in Equation (1), we can obtain the nodal values of the second-order partial derivatives of unknown functions.…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Although nonlinearity, space-time coupling, and high-order derivatives pose major challenges to the accurate and efficient solution of nonlinear wave problems, most recent studies have demonstrated that these difficulties can be resolved very effectively when using the function expansion scheme based on Coiflet-type basis functions [31]. Therefore, for the solution of the nonlinear wave problem shown in Equation (1), we adopted the WICM method [31] in spatial discretization and a multi-step method [32] based on wavelet approximation in time discretization. This processing strategy was expected to allow the resulting algorithm possessing the following advantages: (1) spatial discretization and temporal discretization are completely decoupled; (2) both the spatial discretization and temporal discretization maintain high accuracy consistent with the direct approximation of a function, and almost independent of the order of the equation; (3) nonlinearity can be dealt with in a simple and unified way.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

et al. 2021

Self Cite

View full text Add to dashboard Cite

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

show abstract

“…Following our previous works [24,25,[31][32][33][34], a continuous function f (x) defined on the interval [0, T] can be approximated by the wavelet expansion…”

Section: Wavelet Approximation Of Multiple Integrals In a Bounded Domainmentioning

confidence: 99%

“…The scaling function ϕ(x) and its derivatives ϕ (i) (x) used in Equation ( 6) can be obtained by using the algorithm given in our previous works [32,34].…”

Section: Wavelet Approximation Of Multiple Integrals In a Bounded Domainmentioning

confidence: 99%

Section: Solution Proceduresmentioning

confidence: 99%