Over the past twenty years, the field of plasmonics has been revolutionized with the isolation and utilization of two-dimensional materials, particularly graphene. Consequently, there is significant interest in rapid, robust, and highly accurate computational schemes which can incorporate such materials. Standard volumetric approaches can be contemplated, but these require huge computational resources. Here we describe an algorithm which addresses this issue for nonlocal models of the electromagnetic response of graphene. Our methodology not only approximates the graphene layer with a surface current, but also reformulates the governing volumetric equations in terms of surface quantities using Dirichlet-Neumann Operators. We have recently shown how these surface equations can be numerically simulated in an efficient, stable, and accurate fashion using a High-Order Perturbation of Envelopes methodology. We extend these results to the nonlocal model mentioned above, and using an implementation of this algorithm, we study absorbance spectra of TM polarized plane-waves scattered by a periodic grid of graphene ribbons.

In this paper, a hyperbolic relaxation model is designed for a class of entropy dissipative systems of viscous conservation laws, such as the 1-D viscous Burgers and 2-D Navier-Stokes equations. An artificial variable is introduced to relax both the convective and viscous fluxes together. Based on the entropy dissipative property of the original system, a dissipation condition is proposed for the resulting relaxation model, and used to prove the entropy inequality of the relaxation model for linear convection-diffusion equations. Lax-Wendroff type second-order finite-volume schemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonic speeds, are used to validate the relaxation model and evaluate the performance of the current schemes.

In this paper, we propose an unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element method and the flexibility of the unfitted Nitsche’s method. We also use tailored ghost penalty terms to enhance its robustness. We establish optimal $hp$ convergence rates for both elliptic interface problems and interface eigenvalue problems. Additionally, we demonstrate spectral accuracy for model problems in terms of polynomial degree.

We construct higher order accurate bounds preserving time-implicit Discontinuous Galerkin (DG) discretizations for the reactive Euler equations modelling multispecies and multireaction chemically reactive flows. In numerical discretizations of chemically reactive flows, the time step can be significantly limited because of the large difference between the fluid dynamics time scales and the reaction time scales. In addition, the density and pressure should be nonnegative and the mass fractions between zero and one, which imposes constraints on the numerical solution that must be satisfied to obtain physically reliable solutions. We address these issues using the following steps. Firstly, we develop the Karush-Kuhn-Tucker (KKT) limiter for the chemically reactive Euler equations, which imposes bounds on the numerical solution using Lagrange multipliers, and solve the resulting KKT mixed complementarity problem using a semi-smooth Newton method. The disparity in time scales is addressed using a fractional step method, separating the convection and reaction steps, and the use of higher order accurate Diagonally Implicit Runge-Kutta (DIRK) methods. Finally, Harten’s subcell resolution technique is used to deal with stiff source terms in chemically reactive flows. Numerical results are shown to demonstrate that the bounds preserving KKT-DIRK-DG discretizations are higher order accurate for smooth solutions and able to capture complicated stiff multispecies and multireaction flows with discontinuities.

Accurate and rapid prediction of flowfields is crucial for aerodynamic design. This work proposes a discontinuous Galerkin method (DGM) whose performance can be enhanced with increasing data, for rapid simulation of transonic flow around airfoils under various flow conditions. A lightweight and easily updatable data-driven model is built to predict roughly correct flowfield, and the DGM is then utilized as the CFD solver to refine the detailed flow structures and provide the corrected data. During the construction of the data-driven model, a zonal proper orthogonal decomposition (POD) method is designed to reduce the dimensionality of flowfield while preserving more near-wall flow features, and a weighted distance-based radial basis function (RBF) is constructed to enhance the generalization capability of flowfield prediction. Numerical results demonstrate that the lightweight data-driven model can predict the flowfield around a wide range of airfoils at Mach numbers ranging from 0.7 to 0.95 and angles of attack from $−5^◦$ to $5^◦$ by learning from sparse data, and maintains high accuracy of the location and essential features of flow structures (such as shock waves). In addition, the data-driven model enhanced DGM is able to improve the computational efficiency and simulation robustness as compared to normal DGMs in simulating transonic inviscid/viscous airfoil flowfields on arbitrary grids, and further enables rapid aerodynamic evaluation of numerous sample points during the surrogate-based aerodynamic optimization.

This paper introduces a novel high-order, cell-centered Lagrangian method tailored for curvilinear meshes. To ensure stable curvilinear mesh motions and avoid the algorithmic complexity associated with special mesh stabilization techniques, this method exploits the subcell-wise computational degrees of freedom inherent in the well-established spectral volume (SV) method. It is demonstrated that the numerical results obtained from the SV reconstruction can serve as direct inputs for nodal solvers, yielding favorable mesh node velocities. For cells potentially containing discontinuities, the classical finite volume (FV) reconstruction is applied to each subcell within these cells instead. This subcell-wise reconstruction, analogous to common cell-wise limitations or reconstructions, can effectively reduce unphysical oscillations. More importantly, it is capable of suppressing spurious mesh motions following the same principle of not introducing stabilization techniques such as input correction. The proposed Lagrangian method is completely designed in the computational space with transformed governing equations. Hence, both of the aforementioned SV and FV reconstructions can be performed with the same ease as on uniform and time-independent Eulerian meshes and achieve high-order accuracy. The method further benefits from simplicity and uniformity in its implementation, the compactness of the reconstruction formulae utilized, and the efficiency of the iteration-free nodal solver employed. It exhibits consistently satisfying performance across various numerical tests, including those involving significant mesh deformation and extreme conditions such as low-pressure states accompanied by strong shocks.

In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all-Mach Euler equations with a gravitational source. We start with the conservative form of full compressible Euler equations including conservation of total energy, for shock capturing in the high Mach regime. For asymptotic preserving in the low Mach regime and to address the difficulty of strong coupling between the stiff gravitational source and conservative variables, we add the evolution equation of the perturbation of potential temperature, which corresponds to weak potential temperature stratification under a hydrostatic background potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of potential temperature, we design a novel well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which can be proven to be both asymptotic preserving and asymptotically accurate in the incompressible limit. Numerical experiments are provided to validate these properties.

In this paper, a numerical investigation of power-law fluid flow in the trapezoidal cavity has been conducted by incompressible finite-difference lattice Boltzmann method (IFDLBM). By designing the equilibrium distribution function, the Navier-Stokes equations (NSEs) can be recovered exactly. Through the coordinate transformation method, the body-fitted grid in physical region is transformed into a uniform grid in computational region. The effect of Reynolds $(Re)$ number, the power-law index $n$ and the vertical angle $θ$ on the trapezoidal cavity are investigated. According to the numerical results, we come to some conclusions. For low $Re$ number $Re =100,$ it can be found that the behavior of power-law fluid flow becomes more complicated with the increase of $n.$ And as vertical angle $θ$ decreases, the flow becomes smooth and the number of vortices decreases. For high $Re$ numbers, the flow development becomes more complex, the number and strength of vortices increase. If the Reynolds number increases further, the power-law fluid will changes from steady flow to periodic flow and then to turbulent flow. For the steady flow, the lager the $θ,$ the more complicated the vortices. And the critical $Re$ number from steady to periodic state decreases with the decrease of power-law index $n.$

In this paper, we study a class of second-order accurate and energy-stable numerical schemes for the Cahn-Hilliard model. These schemes are constructed by combining the Crank-Nicolson approximation with three stabilization terms in time and employing the Fourier pseudo-spectral method in space. This class of schemes includes the second-order schemes presented in previous works while providing new schemes by introducing stabilization terms. To solve these schemes with strong nonlinearity efficiently, we propose a linear iteration algorithm and prove that the algorithm satisfies a contraction mapping property in the discrete $l^4$ norm. Furthermore, we establish a comprehensive theoretical analysis, including unique solvability, mass conservation, energy stability, and convergence based on a uniform-in-time $l^∞$ bound of the numerical solution for the proposed second-order scheme. Some numerical simulation results with many different sets of stabilization parameters are presented to conclude the paper.