The new mode of instability found by Tunney et al. [24] is studied with viscous stability theory in this article. When the high-speed boundary layer is subject to certain values of favorable pressure gradient and wall heating, a new mode becomes unstable due to the appearance of the streamwise velocity overshoot ($U(y) > U_∞$) in the base flow. The present study shows that under practical Reynolds numbers, the new mode can hardly co-exist with the conventional first mode and Mack's second mode. Due to the requirement for additional wall heating, the new mode may only lead to laminar-turbulent transition under experimental (artificial) conditions.

In this paper, we propose a Galerkin splitting symplectic (GSS) method for solving the 2D nonlinear Schrödinger equation based on the weak formulation of the equation. First, the model equation is discretized by the Galerkin method in spatial direction by a selected finite element method and the semi-discrete system is rewritten as a finite-dimensional canonical Hamiltonian system. Then the resulted Hamiltonian system is split into a linear Hamiltonian subsystem and a nonlinear subsystem. The linear Hamiltonian subsystem is solved by the implicit midpoint method and the nonlinear subsystem is integrated exactly. By the Strang splitting method, we obtain a fully implicit scheme for the 2D nonlinear Schrödinger equation (NLS), which is symmetric and of order 2 in time. Furthermore, we apply the FFT technique to improve computation efficiency of the new scheme. It is proven that our scheme preserves the mass conservation and the symplectic conservation. Comprehensive numerical experiments are carried out to illustrate the accuracy of the scheme as well as its conservative properties.

In this paper, we combine the Local Method of Approximate Particular Solutions (LMAPS) with polynomial basis functions to solve near-singular problems. Due to the unique feature of the local approach, the LMAPS is capable of capturing the rapid variation of the solution. Polynomial basis functions can become very unstable when the order of the polynomial becomes large. However, since the LMAPS is a local method, the order does not need to be very high; an order of 5 can achieve sufficient accuracy. In order to show the effectiveness of the LMAPS using polynomial basis functions for solving near singular problems, we compare the results with the LMAPS using radial basis functions (RBFs). The advantage of using polynomials as a basis rather than RBFs is that finding an appropriate shape parameter is not necessary.

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully discrete FE approximation possesses the saturation error order under $L^2(Ω)$ norm. Secondly, we theoretically prove the estimation $1+\mathcal{O}(τ^αh^{−2β})$ on the condition number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

The plane wave least-squares method combined with local spectral finite elements has been used effectively to solve time-harmonic acoustic and electromagnetic wave propagation with complex wavenumbers. We develop the plane wave least-squares method and the ultra weak variational formulation for the nonhomogeneous case of the electromagnetic wave propagation in anisotropic media. We derive error estimates of the approximation solutions generated by these methods in one special case of TE mode scattering. Numerical results indicate that the resulting approximate solutions generated by these two methods possess high accuracy and verify the validity of the theoretical results.

In this study, the structural model of a high-aspect-ratio wing is unknown but its structural deformation is measured at some attack angles in a pressured wind tunnel. To implement the static aeroelastic computation at an arbitrary state, an inversion method is proposed to derive the structural stiffness from the known deformation. The wing is simplified into a single-beam model and its bending and torsional flexibility distributions are respectively expressed as a linear combination of several selected basis functions. The bending deformation can be then expressed as a linear combination of the bending deformations of the models structurally characterized by each basis function, which are gradually evaluated by loading the aerodynamic loads computed at the chosen design state. Based on the measured deformation, the bending stiffness distribution is ultimately fitted by a least square method. The torsional stiffness distribution is solved in the same way. Resultantly, a structural deformation-based computational method for static aeroelasticity of a high-aspect-ratio wing model is achieved by combining the structural stiffness inversion method with a coupled computational fluid dynamics (CFD)-computational structural dynamics (CSD) algorithm. The present method is applied to the design and validation states and the numerical results agree well with the experimental data.

This work aims to provide guidance for selection between the implicit and explicit immersed boundary methods (IBMs). The implicit method ensures satisfaction of the boundary condition in one correction step (Wu and Shu, J. Comput. Phys., 228 (2009), pp. 1963–1979). However, it requires formation of a relatively complex matrix. The computational time of this method for moving boundary problems is also a concern, because matrix operations are required at each time step. On the other hand, the explicit multi-direct forcing (MDF) (Luo et al., Phys. Rev. E, 76 (2007), 066709) methods are more straightforward. Nevertheless, the detailed comparison between the two methods is lacked in the literature. In this work, the implicit method, the MDF method, MDF method with under/over-relaxation parameters are compared in detail through simulation of stationary and moving boundary problems. It is found that the implicit method generally outperforms the explicit methods, in respects of maintaining both accuracy and computational efficiency. The present study suggests that it can be worthy to form the matrix and solve the equation system implicitly in IBM for both stationary and moving boundary problems.

In this paper, we present a decoupled finite element scheme for two-dimensional time-dependent viscoelastic fluid flow obeying an Oldroyd-B constitutive equation. The key idea of our decoupled scheme is to divide the full problem into two subproblems, one is the constitutive equation which is stabilized by using discontinuous Galerkin (DG) approximation, and the other is the Stokes problem, can be computed parallel. The decoupled scheme can reduce the computational cost of the numerical simulation and implementation is easy. We compute the velocity $u$ and the pressure $p$ from the Stokes like problem, another unknown stress $σ$ from the constitutive equation. The approximation of stress, velocity and pressure are respectively, $P_1$-discontinuous, $P_2$-continuous, and $P_1$-continuous finite elements. The well-posedness of the finite element scheme is presented and derive the stability analysis of the decoupled algorithm. We obtain the desired error bound also demonstrate the order of the convergence, stability and the flow behavior with the support of two numerical experiments which reveals that decoupled scheme is more efficient than coupled scheme.

Nonconforming $EQ^{rot}_1$ element is applied to solving a kind of nonlinear Benjamin-Bona-Mahony (BBM for short) equation both for space-discrete and fully discrete schemes. A new important estimate is proved, which improves the result of previous works with the exact solution $u$ belonging to $H^2(Ω)$ instead of $H^3(Ω)$. And then, the supercloseness and global superconvergence estimates in broken $H^1$ norm are obtained for space-discrete scheme. Further, the superclose estimates are deduced for backward Euler and Crank-Nicolson schemes. To confirm our theoretical analysis, numerical experiments for backward Euler scheme are executed. It seems that the results presented herein have never been seen for nonconforming FEMs in the existing literature.

The closed-form particular solutions with polynomial basis functions for general partial differential equations (PDEs) with constant coefficients have been derived and applied for solving various kinds of problems in the context of the method of approximate particular solutions (MAPS). In this paper, we propose to extend the above-mentioned method to PDEs with variable coefficients by the substituting and adding-back technique. Since the linear system derived from the polynomial particular solutions is notoriously ill-conditioned, the multiple scale method is applied to alleviate this difficulty. To validate our proposed method, four numerical examples are considered and compared with those obtained by the MAPS using the radial basis functions.

In this paper, air chemical non-equilibrium effects on the shock-induced combustion flow are numerically investigated for a reaction control system (RCS) with gaseous ethylene fuel by solving multi-component Navier-Stokes (N-S) equations. An integrated numerical method is developed that considers two different chemical reaction mechanisms: the high temperature air chemical non-equilibrium reactions and ethylene-oxygen combustion reactions. The method is independently validated by two types of reacting flow: the hypersonic air chemical non-equilibrium flow over a sphere and supersonic ethylene-oxygen combustion flow for a dual combustion chamber. Furthermore, the mixed reacting flow over a blunt cone with a transverse multi-component gaseous jet is analyzed in detail. Numerical results indicate that air chemical non-equilibrium effects could lead to a reduction of the shock detachment distance, a decrease of the temperature behind the shock wave and a reduction of the combustion products.

In this paper, we analyze semi-discrete and fully discrete mixed finite element methods for linear elastodynamics problems in the symmetric formulation. For a large class of conforming mixed finite element methods, the error estimates for each scheme are derived, including the energy norm and $L^2$ norm for stress, and the $L^2$ norm for velocity. All the error estimates are robust for the nearly incompressible materials, in the sense that the constant bound and convergence order are independent of Lamé constant λ. The stress approximation in both norms, as well as the velocity approximation in $L^2$ norm, are with optimal convergence order. Finally numerical experiments are provided to confirm the theoretical analysis.