A new Explicit Immersed Boundary method (IBM) is presented in this work by analyzing and simplifying the system of equations developed from the implicit boundary condition-enforced immersed boundary method. By this way, the requirement to solve the matrix system has been bypassed. It makes the solver be computationally less expensive, especially when large number of Lagrangian points are used to represent the solid boundary. The lattice Boltzmann Flux solver (LBFS) was chosen as the flow solver in this paper as it combines the advantages of both Lattice Boltzmann (LB) solver and Navier-Stokes solver. However, it should be indicated that the new IBM can be incorporated into any flow solver. Comprehensive validations demonstrate that the new explicit scheme bears comparable numerical accuracy as the previous implicit IBM when having a geometry with curvature. The new method is computationally much more efficient than the previous method, especially for moving boundary problems.

In this paper, a linearized difference scheme is proposed for the Sine-Gordon equation (SGE) with a Caputo time derivative of order $\alpha\in(1,2)$. Comparing with the existing linearized difference schemes, the proposed numerical scheme is simpler and easier for theoretical analysis. The solvability, boundedness and convergence of the difference scheme are rigorously established in the $L_{\infty}$ norm. Finally, several numerical experiments are provided to support the theoretical results.

A particle-cluster treecode based on barycentric Lagrange interpolation is presented for fast summation of hydrodynamic interactions through general Rotne-Prager-Yamakawa tensor in 3D. The interpolation nodes are taken to be Chebyshev points of the 2nd kind in each cluster. The barycentric Lagrange interpolation is scale-invariant that promotes the treecode's efficiency. Numerical results show that the treecode CPU time scales like $\mathcal{O}(N \log N)$, where $N$ is the number of beads in the system. The kernel-independent treecode is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.

In this paper, our purpose is to study the unconditional stability and convergence of characteristics finite element method (FEM) for the time-dependent viscoelastic Oldroyd fluids motion equations. We deduce optimal error estimates in $L^2$ and $H^1$ norm. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. Finally, numerical results confirm the theoretical predictions.

In hypersonic boundary layers, Mack mode is the most unstable mode and its secondary instability is a hot research topic on the laminar-turbulent transition. Understanding the mechanism of a secondary instability is very important to delay/promote turbulence generation. In this paper, we focus on the main routes of secondary instability to turbulence in hypersonic flows, including fundamental breakdown and subharmonic breakdown, especially the former one. Through the linear and non-linear stability analyse and secondary instability analysis at various flow temperature conditions, we are trying to find out the temperature effect on the secondary instability mechanism of Mack mode disturbances. The results point out that the fundamental mode always dominates the breakdown type when the saturated amplitude of the primary Mack mode is large enough. As the stagnation temperature increases, the maximum growth-rates of the fundamental mode and subharmonic mode both increase. Meanwhile, when the wall is cooling, the maximum growth-rates of the fundamental mode and subharmonic mode are both enlarged. In contrast, with the heating wall, the maximum growth-rates of the secondary instability both decrease.

We mainly research the reduced-order of the classical natural boundary element (CNBE) method for the two-dimensional (2D) viscoelastic wave equation by means of proper orthogonal decomposition (POD) technique. For this purpose, we firstly establish the CNBE model and analyze the existence, stability, and errors for the CNBE solutions. We then build a highly efficient reduced-order extrapolating natural boundary element (HEROENBE) mode including few degrees of freedom but possessing sufficiently high accuracy for the 2D viscoelastic wave equation by the POD method and analyze the existence, stability, and errors of the HEROENBE solutions by the CNBE method. We finally employ some numerical experiments to verify that the numerical results are accorded with the theoretical ones so that the validity for the HEROENBE model is further verified.

Two adaptive techniques for choosing relaxation factor, namely, Minimal Residual Relaxation (MRR) and Orthogonal Projection Relaxation (OPR), on basic iterative methods for solving linear systems are proposed. Unlike classic relaxation, in which the optimal relaxation factor is generally difficult to find, in these proposed techniques, non-stationary relaxation factor based on minimal residual or orthogonal projection method is calculated adaptively in each relaxation step with acceptable cost for Jacobi, Gauss-Seidel or symmetric Gauss-Seidel iterative methods. In order to avoid the "stagnation" of the successive locally optimal relaxations, a recipe of inserting several basic iterations between every two adjacent relaxations is suggested and the resulting MRR$(m)$/OPR$(m)$ strategy is more stable and efficient (here $m$ denotes the number of basic iterations inserted). To solve linear systems with multiple right-hand sides efficiently, block-form relaxation strategies are proposed based on the MRR$(m)$ and OPR$(m)$. Numerical experiments show that the presented MRR$(m)$/OPR$(m)$ algorithm is more robust and effective than classic relaxation methods. It is also showed that the proposed block relaxation strategies can efficiently accelerate the solution of systems with multiple right-hand sides in terms of total solution time as well as number of iterations.

In this paper, we study an implicit-explicit scheme for the radiation hydrodynamics in the equilibrium diffusion limit and in the grey nonequilibrium diffusion limit. We extend a popular Godunov-type method, the MUSCL-Hancock scheme, to the convective part of the radiation hydrodynamics, while a cell centered finite volume scheme is used for the radiative heat transfer. Moreover, the implicit-explicit scheme is easier to implement. Numerical simulations show the character of the radiative shock wave and the accuracy of the scheme.

In this study, an improved single-relaxation-time multiphase lattice Boltzmann method (SRT-MLBM) is developed for simulating multiphase flows with both large density ratios and high Reynolds numbers. This model employs two distribution functions in lattice Boltzmann equation (LBE), with one tracking the interface between different fluids and the other calculating hydrodynamic properties. In the interface distribution function, a time derivative term is introduced to recover the Cahn-Hilliard equation. For flow field, a modified equilibrium particle distribution function is present to evolve the velocity and pressure field. The present method keeps simplicity of the conventional SRT-MLBM but enjoys good stability property in simulating multiphase. Apart from several benchmarks, the present model is validated by simulating various challenging multiphase flows, including two droplets impact on liquid film, droplet oblique splashing on a thin film and a drop impact on a moving liquid film. Numerical results show the reliability of present model for effectively simulating complex multiphase flows at density ratios of 1000 and high Reynolds numbers (up to 7000).

A novel upwind technique for the localized method of approximate particular solutions (LMAPS) is proposed to solve the convection-diffusion equations. An upwind approximation to the convective terms is implemented by choosing upwind interpolation stencils while the central interpolation stencils are used for the diffusive terms. The proposed upwind LMAPS scheme is also been compared with conventional LMAPS without upwind technique, to demonstrate its superiority in generating high accurate solutions than the latter. Numerical results show that the proposed upwind LMAPS has high accuracy and efficiency for a variety of convection-diffusion equations.

The prediction and control of the laminar-turbulent transition in three-dimensional boundary layers are crucial for the designs of vehicles, aircrafts, etc. Receptivity, the initial stage of transition, is the key to implement the prediction and control of the transition process. Former experimental results showed that, under a relatively low-level turbulence, the three-dimensional boundary-layer transition is mainly induced by the stationary cross-flow modes rather than the travelling ones. Near the leading edge, stationary cross-flow modes can be excited by three-dimensional localized roughness. And the receptive process is affected by both the size of roughness and the shape of leading edge which distorts the mean flow over the plate. Therefore, we perform direct numerical simulation to investigate the excitation of stationary cross-flow modes by three-dimensional localized wall roughness in swept-plate boundary layers with various elliptic leading edges. The effect of the leading-edge curvature on the induced stationary cross-flow modes is revealed. And the relations of the leading-edge curvatures with the amplitudes and dispersion relations of the stationary cross-flow modes are determined. Furthermore, the correlations between the receptivity coefficients and the geometries, locations and numbers of the roughness respectively are analyzed in different leading-edge curvatures. This research aims to complement the study of receptivity in the cross-flow boundary layer.

In this paper, we develop the superconvergence analysis of two-grid algorithm by Crank-Nicolson finite element discrete scheme with the lowest Nédélec element for nonlinear power-law conductivity in Maxwell's problems. Our main contribution will have two parts. On the one hand, in order to overcome the difficulty of misconvergence of classical two-grid method by the lowest Nédélec element, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the numerical solution on the coarse mesh classically. On the other hand, we push the two-grid solution to high accuracy by the postprocessing interpolation technique. Such a design can improve the computational accuracy in space and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate $\mathcal{O}(\Delta t^2+h^2+H^{\frac{5}{2}})$ in three-dimension space, which means that the space mesh size satisfies $h=\mathcal{O}(H^\frac{5}{4})$. We also present two examples to verify our theorem.