In this article a sixth-order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth-order finite difference approximation scheme for a two-point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels-Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second-order Crank-Nicolson scheme as well as Sun-Zhang's recent fourth-order method.

No abstract

In this work, flow separation control has been conducted for the S809 aerofoil at a high Reynolds number using synthetic jet technology. The aerodynamic characteristics of the aerofoil have been compared in detail at different angles of attack, for the cases with and without adoption of synthetic jet. Numerical methods are employed for predicting flow structure and performance of the aerofoil. In addition, main parameters of the synthetic jet are optimized by the orthogonal experimental design, and dual jets are also employed for the comparison to a single jet. The results show that the flow separation at large angles of attack can be eliminated or greatly reduced by the synthetic jet, due to the mixing of low-energy fluid in boundary layer with high-energy fluid produced by the synthetic jet. The lift-to-drag ratio has been considerably increased by the synthetic jet for the critical condition, deep stall condition and complete stall condition as well. The maximum jet velocity of the synthetic jet is found to have the biggest effect on flow separation control. Furthermore, compared with single synthetic jet, the dual jets can make much better improvement on flow separation control of the aerofoil, especially at the complete stall condition.

No abstract

This paper is devoted to studying the following nonlinear fractional problem: $$ \textstyle\begin{cases} (-\Delta )^{s}u+u=K( \vert x \vert )u^{p},\quad u>0, x\in {\mathbb{R}}^{N}, \\ u(x)\in H^{s}({\mathbb{R}}^{N}), \end{cases} $$ { ( − Δ ) s u + u = K ( | x | ) u p , u > 0 , x ∈ R N , u ( x ) ∈ H s ( R N ) , where $N\geq 3$ N ≥ 3 , $0< s<1$ 0 < s < 1 , $1< p<\frac{N+2s}{N-2s}$ 1 < p < N + 2 s N − 2 s , $K(|x|)$ K ( | x | ) is a positive radical function. We constructed infinitely many non-radial solutions of the new type which have a more complex concentration structure for (0.1).