Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...
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Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping *** resulting iterative schemes have a fast convergence rate to steady-state ***,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local ***,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the *** multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational *** this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational ***,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)*** experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the gr...
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Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou＇s direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.
In secure multi-party computations (SMC), parties wish to compute a function on their private data without revealing more information about their data than what the function reveals. In this paper, we investigate two ...
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High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th...
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High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional *** our previous work(Lu et *** Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse *** this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO *** experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
In this paper, we investigate the dynamical instability of the dark state in the conversion of Bose-Fermi mixtures into stable molecules through a stimulated Raman adiabatic passage aided by Feshbach resonance. We ana...
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In this paper, we investigate the dynamical instability of the dark state in the conversion of Bose-Fermi mixtures into stable molecules through a stimulated Raman adiabatic passage aided by Feshbach resonance. We analytically obtain the regions where the dynamical instability appears and find that such instability in the Bose-Fermi mixture system is caused not only by bosonic interparticle interactions but also by Pauli blocking terms, which is different from the scenario of a pure bosonic system where instability is induced by nonlinear interparticle collisions. Taking a 40K-87Rb mixture as an example, we give the unstable regions numerically.
J. Chem. Soc., Faraday Trans. 2, 1987, 83, 949-959 Numerical Solution of a Poisson-Boltzmann Theory for a Primitive Model Electrolyte with Size and Charge Asymmetric Ions Christopher W. Outhwaite department of applied...
J. Chem. Soc., Faraday Trans. 2, 1987, 83, 949-959 Numerical Solution of a Poisson-Boltzmann Theory for a Primitive Model Electrolyte with Size and Charge Asymmetric Ions Christopher W. Outhwaite department of applied and computationalmathematics, fie University, Shefield S102TN A numerical solution is given of a Poisson-Boltzmann theory with a sym- metric radial distribution function. Comparisons are made with the HNC integral equation for 1:2 and 1:3 electrolytes and with 1 :1 and 2: 2 Monte Carlo results for different ion sizes. Some results are also given for 1:2,2 :1 and 1:3, 3 :1 electrolytes with unequal ion sizes. The Poisson-Boltzmann (PB) equation has for many years played a major role in the interpretation of the thermodynamic and structural properties of electrolyte solution^.'-^ During the last 15 years its role has been to a large extent superceded by integral equation techniques, although it still continues to attract attenfi~n.~-~ Indeed it is relatively simple to solve numerically and it gives a fairly accurate picture of 1 : 1 and to a lesser extent 2 :2 restricted primitive model electrolytes at the lower concentrations.A serious draw- back to its wider use is that the PB radial distribution function gU(r)for two ions of species i and j is inconsistent when the ion species are of different valences or have different ionic radii. We present here some numerical calculations for a single electrolyte with a symmetric gii(r) which depends on the solution of a pair of coupled equations for the mean electrostatic potentials t,hi(r) and t,hj(r).The results indicate that this symmetric PB theory is a viable theory for primitive model electrolytes at the lower electrolyte concentrations in those situations where the standard PB gii(r) is inconsistent. Theory Ftat and Levine,’ using the Kirkwood charging process on two ions i and j, derived a symmetric distribution function g, in the diffuse double layer. Adapting this theory to a primitive model electrolyt
Compressive sensing(CS)is an emerging methodology in computational signal processing that has recently attracted intensive research *** present,the basic CS theory includes recoverability and stability:the former quan...
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Compressive sensing(CS)is an emerging methodology in computational signal processing that has recently attracted intensive research *** present,the basic CS theory includes recoverability and stability:the former quantifies the central fact that a sparse signal of length n can be exactly recovered from far fewer than n measurements via l1-minimization or other recovery techniques,while the latter specifies the stability of a recovery technique in the presence of measurement errors and inexact *** far,most analyses in CS rely heavily on the Restricted Isometry Property(RIP)for *** this paper,we present an alternative,non-RIP analysis for CS via *** purpose is three-fold:(a)to introduce an elementary and RIP-free treatment of the basic CS theory;(b)to extend the current recoverability and stability results so that prior knowledge can be utilized to enhance recovery via l1-minimization;and(c)to substantiate a property called uniform recoverability of l1-minimization;that is,for almost all random measurement matrices recoverability is asymptotically *** the aid of two classic results,the non-RIP approach enables us to quickly derive from scratch all basic results for the extended theory.
This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (P...
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This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (PEC) boundary condition, employing the energy method. To this end, we first establish some new energy identities for the 2D Maxwell equations in a lossy medium with a PEC boundary condition. Then by making use of these energy identities, it is proved that the FDTD scheme and its time difference scheme are stable in the discrete L2 and H1 norms when the CFL condition is satisfied. It is shown further that the solution to both the FDTD scheme and its time difference scheme is second-order convergent in both space and time in the discrete L2 and H1 norms under a slightly stricter condition than the CFL condition. This means that the solution to the FDTD scheme is superconvergent. Numerical results are also provided to confirm the theoretical analysis.
Quantum computing, a prominent non-Von Neumann paradigm beyond Moore’s law, can offer superpolynomial speedups for certain problems. Yet its advantages in efficiency for tasks like machine learning remain under inves...
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