We study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bénard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these snapshots as column vectors. The singular values decay rapidly and allow us to extract a small percentage of left singular vectors corresponding to the top singular values as adaptive basis vectors. The trained orthogonal adaptive basis makes possible low cost computation of the effective diffusivities at smaller molecular diffusivities (testing). The testing errors decrease as the training occurs at smaller molecular diffusivities. We make use of the Poincaré map of the advection-diffusion equation to bypass long time simulation and gain accuracy in computing effective diffusivity and learning adaptive basis. We observe a non-monotone relationship between residual diffusivity and the amount of chaos in the advection, though the overall trend is that sufficient chaos leads to higher residual diffusivity.

We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

We propose a numerical procedure to extend to full aperture the acoustic far-field pattern (FFP) when measured in only few observation angles. The reconstruction procedure is a multi-step technique that combines a total variation regularized iterative method with the standard Tikhonov regularized pseudo-inversion. The proposed approach distinguishes itself from existing solution methodologies by using an exact representation of the total variation which is crucial for the stability and robustness of Newton algorithms. We present numerical results in the case of two-dimensional acoustic scattering problems to illustrate the potential of the proposed procedure for reconstructing the full aperture of the FFP from very few noisy data such as backscattering synthetic measurements.

The theory of reproducing kernel Hilbert spaces is applied to a minimization problem with prescribed nodes. We re-prove and generalize some results previously obtained by Gunawan et al. [2,3], and also discuss the Hölder continuity of the solution to the problem.

A high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.