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We formulate hydrodynamic equations and spectrally accurate numerical methodsfor investigating the role of geometry in flows within two-dimensional fluidinterfaces. To achieve numerical approximations having high precision and levelof symmetry for radial manifold shapes, we develop spectral Galerkin methodsbased on hyperinterpolation with Lebedev quadratures for $L^2$-projection tospherical harmonics. We demonstrate our methods by investigating hydrodynamicresponses as the surface geometry is varied. Relative to the case of a sphere,we find significant changes can occur in the observed hydrodynamic flowresponses as exhibited by quantitative and topological transitions in thestructure of the flow. We present numerical results based on theRayleigh-Dissipation principle to gain further insights into these flowresponses. We investigate the roles played by the geometry especiallyconcerning the positive and negative Gaussian curvature of the interface. Weprovide general approaches for taking geometric effects into account forinvestigations of hydrodynamic phenomena within curved fluid interfaces.

A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over bounded volumes that have smooth boundaries in three dimensions is described. A key aspect of this approach is that it allows the user to approximate the value of the integral without explicit knowledge of an expression for the boundary surface. Instead, a tesselation of the node set is utilized to inform the algorithm of the domain geometry. Further, the method applies to node sets featuring spatially varying density, facilitating its use in Applied Mathematics, Mathematical Physics and myriad other application areas, where the locations of the nodes might be fixed by experiment or previous simulation. By using the RBF-FD-like approach, the proposed algorithm computes quadrature weights for N arbitrarily scattered nodes in only O (N log ⁡ N) operations with tunable orders of accuracy. • A fast and accurate method for quadrature over 3D volumes with smooth boundaries. • High orders of accuracy are achieved on node sets featuring no particular uniformity. • Can be distributed to any number of available processors. • Knowledge of an expression for the boundary surface is unnecessary. [ABSTRACT FROM AUTHOR]

The capability to rapidly and robustly simulate the scattering of linear waves by periodic, multiply layered media in two and three dimensions is crucial in many engineering applications. In this regard, we present a High-Order Perturbation of Surfaces method for linear wave scattering in a multiply layered periodic medium to find an accurate numerical solution of the governing Helmholtz equations. For this we truncate the bi-infinite computational domain to a finite one with artificial boundaries, above and below the structure, and enforce transparent boundary conditions there via Dirichlet–Neumann Operators. This is followed by a Transformed Field Expansion resulting in a Fourier collocation, Legendre–Galerkin, Taylor series method for solving the problem in a transformed set of coordinates. Assorted numerical simulations display the spectral convergence of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method. [ABSTRACT FROM AUTHOR]

We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes slowly with time. In the later case, the evolution operator can be effectively approximated in a step-by-step manner: first divide the time integration interval in sufficiently short subintervals, and then successively solve a Schrödinger equation with a different time-independent Hamiltonian operator in each of these subintervals. When discretized in space, the Schrödinger equation with the time-independent Hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous Hamiltonian system corresponding to a system of coupled harmonic oscillators. The particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals. Sharp local error bounds are obtained for the solution of the linear autonomous Hamiltonian system considered in each time subinterval. Our schemes can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments performed for different time-independent Hamiltonians, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time interval lengths. The algorithm we present automatically selects, for each time subinterval, the most efficient splitting scheme (among several new optimized splitting methods) for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the Hamiltonian operator. [ABSTRACT FROM AUTHOR]