We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling. The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy. A detailed discussion of algorithmic details is given, as well as proofs of the claimed properties. We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative. This leads to numerical difficulties, which seem to have discouraged further development of complex behavioral continuum models, which is a significant research need.
In this paper, we develop a robust numerical scheme that solves hyperbolic traffic flow models based on their non-conservative form. We develop a fifth-order alternative weighted essentially non-oscillatory (A-WENO) finite-difference scheme based on the path-conservative central-upwind (PCCU) method for several non-equilibrium traffic flow models. In order to treat the non-conservative product terms, we use a path-conservative technique. To this end, we first apply the recently proposed second-order finite-volume PCCU scheme to the traffic flow models, and then extend this scheme to the fifth-order of accuracy via the finite-difference A-WENO framework. The designed schemes are applied to three different traffic flow models and tested on a number of challenging numerical examples. Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher. The proposed scheme in this paper sets the stage for developing more robust and complex continuum traffic flow models with respect to human psychological factors.

Bulk-surface partial differential equations (BS-PDEs) are prevalent in many applications such as cellular, developmental and plant biology as well as in engineering and material sciences. Novel numerical methods for BS-PDEs in three space dimensions (3D) are sparse. In this work, we present a bulk-surface virtual element method (BS-VEM) for bulk-surface reaction-diffusion systems, a form of semilinear parabolic BS-PDEs in 3D. Unlike previous studies in two space dimensions (2D), the 3D bulk is approximated with general polyhedra, whose outer faces constitute a flat polygonal approximation of the surface. For this reason, the method is restricted to the lowest order case where the geometric error is not dominant. The BS-VEM guarantees all the advantages of polyhedral methods such as easy mesh generation and fast matrix assembly on general geometries. Such advantages are much more relevant than in 2D. Despite allowing for general polyhedra, general nonlinear reaction kinetics and general surface curvature, the method only relies on nodal values without needing additional evaluations usually associated with the quadrature of general reaction kinetics. This latter is particularly costly in 3D. The BS-VEM as implemented in this study retains optimal convergence of second order in space.

This paper presents a fourth-order Cartesian grid based boundary integral method (BIM) for heterogeneous interface problems in two and three dimensional space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the governing equation with interface conditions into boundary integral equations (BIEs) and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region. Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver. The structure of the coefficient matrix is preserved even with the existence of the interface. In the whole calculation process, analytical expressions of Green’s functions are never determined, formulated or computed. This is the novelty of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.

Blind deblurring for color images has long been a challenging computer vision task. The intrinsic color structures within image channels have typically been disregarded in many excellent works. We investigate employing regularizations in the hue, saturation, and value (HSV) color space via the quaternion framework in order to better retain the internal relationship among the multiple channels and reduce color distortions and color artifacts. We observe that a geometric spatial-feature prior utilized in the intermediate latent image successfully enhances the kernel accuracy for the blind deblurring variational models, preserving the salient edges while decreasing the unfavorable structures. Motivated by this, we develop a saturation-value geometric spatial-feature prior in the HSV color space via the quaternion framework for blind color image deblurring, which facilitates blur kernel estimation. An alternating optimization strategy combined with a primal-dual projected gradient method can effectively solve this novel proposed model. Extensive experimental results show that our model outperforms state-of-the-art methods in blind color image deblurring by a wide margin, demonstrating the effectiveness of the proposed model.

We propose a clustering-based approach for identifying coherent flow structures in continuous dynamical systems. We first treat a particle trajectory over a finite time interval as a high-dimensional data point and then cluster these data from different initial locations into groups. The method then uses the normalized standard deviation or mean absolute deviation to quantify the deformation. Unlike the usual finite-time Lyapunov exponent (FTLE), the proposed algorithm considers the complete traveling history of the particles. We also suggest two extensions of the method. To improve the computational efficiency, we develop an adaptive approach that constructs different subsamples of the whole particle trajectory based on a finite time interval. To start the computation in parallel to the flow trajectory data collection, we also develop an on-the-fly approach to improve the solution as we continue to provide more measurements for the algorithm. The method can efficiently compute the WCVE over a different time interval by modifying the available data points.

For compressible reactive flows with stiff source terms, a new block-based adaptive multi-resolution method coupled with the adaptive multi-resolution representation model for ZND detonation and a conservative front capturing method based on a level-set technique is presented. When simulating stiff reactive flows, underresolution in space and time can lead to incorrect propagation speeds of discontinuities, and numerical dissipation makes it impossible for traditional shock-capturing methods to locate the detonation front. To solve these challenges, the proposed method leverages an adaptive multi-resolution representation model to separate the scales of the reaction from those of fluid dynamics, achieving both high-resolution solutions and high efficiency. A level set technique is used to capture the detonation front sharply and reduce errors due to the inaccurate prediction of detonation speed. In order to ensure conservation, a conservative modified finite volume scheme is implemented, and the front transition fluxes are calculated by considering a Riemann problem. A series of numerical examples of stiff detonation simulations are performed to illustrate that the present method can acquire the correct propagation speed and accurately capture the sharp detonation front. Comparative numerical results also validate the approach’s benefits and excellent performance.

In this paper, we propose a Newton iterative algorithm to numerically reconstruct a locally rough surface with Dirichlet and impedance boundary conditions by near-field measurements of acoustic waves. The algorithm relies on the Fréchet differentiability analysis of the locally rough surface scattering problem, which is established by reducing the original model into an equivalent boundary value problem with compactly supported boundary data. With a slight modification, the algorithm can be also extended to reconstruct the local perturbation of a non-local rough surface. Finally, numerical results are presented to illustrate the effectiveness of the inversion algorithm with the multi-frequency data.

We show that the lattice Boltzmann method (LBM) based color-gradient model with a central moments formulation (CG-CM) is capable of accurately simulating the droplet-on-demand inkjetting process on a micrometer length scale by comparing it to the Arbitrary Lagrangian Eulerian Finite Element Method (ALE-FEM). A full jetting cycle is simulated using both CG-CM and ALE-FEM and results are quantitatively compared by measuring the ejected ink velocity, volume and contraction rate. We also show that the individual relevant physical phenomena are accurately captured by considering three test-cases; droplet oscillation, ligament contraction and capillary rise. The first two cases test accuracy for a dynamic system where surface tension is the driving force and the third case is designed to test wetting boundary conditions. For the first two cases we also compare the CG-CM and ALE-FEM results to Volume of Fluid (VOF) simulations. Comparison of the three methods shows close agreement when compared to each other and analytical solutions, where available. Finally we demonstrate that asymmetric jetting is achievable using 3D CG-CM simulations utilizing asymmetric wetting conditions inside the jet-nozzle. This allows for systematic investigation into the physics of asymmetric jetting, e.g. due to jet-nozzle manufacturing imperfections or due to other disturbances.