This article is devoted to the development of a new approach for stress recovery in the isogeometric analysis method which makes use of equilibrium of patches. It is shown that the obtained stresses of this approach are more accurate than our previous work based on using the super convergent property of the Gauss quadrature points. To demonstrate the efficiency of this error estimator, a few examples of elasticity problems with available analytical solutions are ‎modeled and solved. The obtained results are ‎compared with the exact solutions as well as our ‎previous method. These ‎results indicate superiority of the current method that can be considered ‎as a simple and ‎efficient approach for stress recovery and error estimation in the isogeometric ‎analysis ‎method.

In this paper, a kernel based method, the modified localized method of approximated particular solutions (MLMAPS) [16, 23] is utilized to solve unsteady-state linear and nonlinear diffusion-reaction PDEs with or without convections. The time-space and spatial space are discretized by the higher-order Houbolt method with various time step sizes and the MLMAPS, respectively. The local truncation error associated with the time discretization is $\mathcal{O}(h^4)$, where $h$ is the largest time step size used. The spatial domain is then treated by a special kernel, the integrated polyharmonic splines kernels together with low-order polynomial basis. Typical computational algorithms require a trade off between accuracy and rate of convergency. However, the experimental analysis has shown high accuracy and fast convergence of the proposed method.

The convective heat transfer from two tandemly arranged circular cylinders, with the downstream one subjected to controlled transverse oscillation, is numerically studied in this work. At Reynolds number 100, eight oscillation frequencies varying from 0.4 to 1.7 and two oscillation amplitudes of 0.15 and 0.35 are examined at three spacing distances, which are corresponding to the Vortex Suppression, critical and the Vortex Formation regimes. Results show that in such a system, the spacing distance determines the flow regime and set the base for the flow patterns, the forcing and heat transfer characteristics. For the flow features, the locked-on phenomenon is observed in the present study, whereby the flow and thermal fields are phase-locked with the cylinder oscillation. Outside the locked-on region, an ordered and modulated near-wake can also be observed, but the variation period of the flow field is different from the oscillation period. Additionally, it is found that the parameter range to obtain a locked-on response is narrowed with an increasing spacing distance. Compared with the system with two stationary cylinder, the lift and drag force can subject to larger variations (especially for the downstream cylinder) due to the oscillatory motion. More importantly, the present study demonstrates a large room to enhance the heat transfer. It shows that in the Vortex Suppression and the Vortex Formation regimes considered here, the heat transfer rate of the downstream cylinder can be improved in most cases (with that of the upstream cylinder slightly increased or kept the same with the baseline). A maximum increase of 59.4% of the average Nusselt number can be obtained in the test matrix studied.

This paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time $\text{H}^1$-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.

An upwind mixed finite element method is proposed on changing meshes for solving a positive semi-definite miscible displacement problem of Darcy-Forchheimer flow in three-dimensional porous media. The pressure and velocity could be obtained together by using a mixed finite element, and the computational accuracy of velocity is improved. The concentration equation is solved by the upwind mixed finite element scheme on changing meshes, where the upwind approximation and an expanded mixed finite element are adopted for the convection and diffusion, respectively. It solves the convection-dominated diffusion problem well and has the following improvements. First, the conservation of mass, an important physical nature, is preserved. Second, it has high order computational accuracy. An optimal-order error estimates is concluded. Numerical experiments illustrate the efficiency and application of the presented scheme.

We generalize the troubled-cell indicator on unstructured triangular meshes recently introduced by Fu and Shu (J. Comput. Phys., 347 (2017), pp. 305--327) to $h$-adaptive rectangular meshes where hanging nodes exist. The generalized troubled-cell indicator keeps the good properties of simplicity, compactness and insensitivity to particular test cases. Numerical tests on the two-dimensional scalar Burgers' equation and hyperbolic systems of Euler equations demonstrate the good performance of the generalized indicator. The results on both uniform and $h$-adaptive meshes indicate that the generalized indicator is able to capture shocks effectively without any PDE-sensitive parameter to tune.

This paper intends to explore the acceleration performances of parallel algorithms on CPU and GPU devices for two-dimensional Unified Gas-Kinetic Scheme (UGKS) in a comparative way. Such a comparison could provide guidance when the UGKS needs to be accelerated. To accomplish this goal, parallel algorithms on CPU and GPU devices are implemented and the speedup performances are investigated by a case of two-dimensional channel flow. Based on the multiscale feature of UGKS, a two-level fine-grain parallel strategy for both spatial and velocity spaces is adopted for GPU algorithm. The parallel CPU algorithm applies a two dimensional block layout that also parallelizes the spatial and velocity coordinates. A series of meshes with different sizes are tested to reveal the performance evolution of the two algorithms. Special attention is paid to cases where the discrete velocity space is large. The comparisons show that the proposed fine-grain GPU algorithm could take advantage of the feature of UGKS and provides significant speedups especially with the latest GPU version. On the other hand, parallel CPU strategy might provide more predictable and preferable performances when velocity space grid point number is large.

In this paper, we construct a compact difference scheme for the time-fractional partial integro-differential equation. This model involves two nonlocal terms in time, i.e., a Caputo time-fractional derivative and an integral term with memory. We obtain the stability and the discrete $L_{2}$ convergence with second-order in time and fourth-order in space by the energy method. Two numerical examples are provided to confirm the theoretical results.

In this paper, we analytically derived the degenerate scale of an infinite plane containing two unequal circles and numerically implemented by using the BEM. We provide two methods to analytically derive the degenerate scale. One is using the degenerate kernel and the other is using the conformal mapping. The closed-form fundamental solution of the two-dimensional Laplace equation is expanded to the degenerate kernel form in order to analytically study the degenerate scale in the BIE. Moreover, we used the technique of the conformal mapping in order to analytically study the degenerate scale in the complex variables. Then, a boundary value problem can be transformed into a Green's function. Finally, we prove the equivalence of the two analytical formulas derived by using the degenerate kernel and by using the complex variables. They are also examined by using the BEM. Good agreement is made. Besides, the case of the two equal circles is just a special one of the present formula.

Portfolio credit derivatives, including the basket credit default swaps, are designed to facilitate the transfer of credit risk amongst market participants. Investors consider them as cheap tools to hedge a portfolio of credits, instead of individual hedging of the credits. The prime aim of this work is to model the hazard rate process using stochastic default intensity models, as well as extend the results to the pricing of basket default swaps. We focused on the  $n$th-to-default swaps whereby the spreads are dependent on the $n$th default time, and we estimated the joint survival probability distribution functions of the intensity models under the risk-neutral pricing measure, for both the homogeneous and the heterogeneous portfolio. This work further employed the Monte-Carlo method, under the one-factor Gaussian copula model to numerically approximate the distribution function of the default time, and thus, the numerical experiments for pricing the $n$th default swaps were made viable under the two portfolio types. Finally, we compared the effects of different swap parameters to various $n$th-to-default swaps.