For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples,strong wave reflection occurs under the Bragg resonance condition.The critical wave frequency,at which the ...For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples,strong wave reflection occurs under the Bragg resonance condition.The critical wave frequency,at which the peak reflection coefficient is obtained,has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction.It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects,but the intrinsic mechanism remains unclear.By a regular perturbation analysis,we derive the theoretical solution of frequency downshift due to third-order nonlinear effects of both bottom and free-surface boundaries.It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actually causes frequency upshift.The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness.Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom.In addition,direct numerical simulations based on the high-order spectral method are conducted to validate the present theory.The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.展开更多
An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained ex...An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis.And a sharp estimate on the algebraic system's condition number is established which behaves as N4s with respect to the polynomial degree N,where 2s is the fractional derivative order.The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces.Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived.Meanwhile,rigorous error estimates of the eigenvalues and eigenvectors are ob-tained.Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.展开更多
Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optim...Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optimal convergence rates are proved.展开更多
基金financially supported by the National Natural Science Foundation of China (Grant Nos. U1706230 and51379071)the Key Project of NSFC-Shandong Joint Research Funding POW3C (Grant No. U1906230)the National Science Fund for Distinguished Young Scholars (Grant No. 51425901)
文摘For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples,strong wave reflection occurs under the Bragg resonance condition.The critical wave frequency,at which the peak reflection coefficient is obtained,has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction.It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects,but the intrinsic mechanism remains unclear.By a regular perturbation analysis,we derive the theoretical solution of frequency downshift due to third-order nonlinear effects of both bottom and free-surface boundaries.It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actually causes frequency upshift.The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness.Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom.In addition,direct numerical simulations based on the high-order spectral method are conducted to validate the present theory.The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.
基金supported by the National Natural Science Foundation of China(Grant No.12101325)and by the NUPTSF(Grant No.NY220162)The second author was supported by the National Natural Science Foundation of China(Grant Nos.12131005,11971016)+1 种基金The third author was supported by the National Natural Science Foundation of China(Grant No.12131005)The fifth author was supported by the National Natural Science Foundation of China(Grant Nos.12131005,U2230402).
文摘An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper.The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis.And a sharp estimate on the algebraic system's condition number is established which behaves as N4s with respect to the polynomial degree N,where 2s is the fractional derivative order.The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces.Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived.Meanwhile,rigorous error estimates of the eigenvalues and eigenvectors are ob-tained.Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.
基金The work of the first author was supported in part by Yunnan Provincial Science and Technology Department Research Award:Interdisciplinary Research in Computational Mathematics and Mechanics with Applications in Energy Engineering(Grant No.2009CI128)Yunnan Provincial Science and Technology Project(Grant No.2014RA071)The work of the second author was supported partially by the National Natural Science Foundation of China with Grant Nos 11471026 and 11871465 and National Centre for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences.
文摘Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optimal convergence rates are proved.
