This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in...This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.展开更多
We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into ...We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].展开更多
In this work,we introduce a kind of new structured radial grating,which is named the even-type sinusoidal amplitude radial(ETASR)grating.Based on diffraction theory and the principle of stationary phase,a comprehensiv...In this work,we introduce a kind of new structured radial grating,which is named the even-type sinusoidal amplitude radial(ETASR)grating.Based on diffraction theory and the principle of stationary phase,a comprehensive theoretical investigation on the diffraction patterns of ETASR gratings is conducted.Theoretical results show that novel carpet beams with beautiful optical structures and distinctive characteristics have been constructed on the basics of the ETASR grating.Their diffraction patterns are independent of propagation distance,that is,the new carpet beams have diffraction-free propagating characteristics.The non-diffracting carpet beams are divided into two types by beam characteristics:non-diffracting integer-order and half-integer-order carpet beams.Subsequently,we experimentally generate these carpet beams using the ETASR grating.Finally,their particularly interesting optical morphology and features are explored through numerical simulations and experiments.展开更多
文摘This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.
基金supported in part by the Knowledge Innovation Project of the Chinese Academy of Sciences Nos. K5501312S1 and K5502212F1, and NSFC grant No. 10601062supported in part by NSF grant Nos. DMS-0305081 and DMS-0608720, NSFC grant No. 10228101 and NSAF grant No. 10676017
文摘We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].
基金supported by the National Natural Science Foundation of China(Nos.11974314 and 11674288).
文摘In this work,we introduce a kind of new structured radial grating,which is named the even-type sinusoidal amplitude radial(ETASR)grating.Based on diffraction theory and the principle of stationary phase,a comprehensive theoretical investigation on the diffraction patterns of ETASR gratings is conducted.Theoretical results show that novel carpet beams with beautiful optical structures and distinctive characteristics have been constructed on the basics of the ETASR grating.Their diffraction patterns are independent of propagation distance,that is,the new carpet beams have diffraction-free propagating characteristics.The non-diffracting carpet beams are divided into two types by beam characteristics:non-diffracting integer-order and half-integer-order carpet beams.Subsequently,we experimentally generate these carpet beams using the ETASR grating.Finally,their particularly interesting optical morphology and features are explored through numerical simulations and experiments.
