In this paper, we study the Jacobi frame approximation with equispaced samples and derive an error estimation. We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter $\gamma$ increases in the uniform norm, especially for differentiable functions. In addition, we show that when the indexes of Jacobi polynomials $α$ and $β$ are larger (for example max$\{α,β\} > 10$), it leads to a divergence behavior on the frame approximation error decay.

A new form of universal wall function is proposed in this work, which can be applied to a variety of eddy viscosity models including one-equation models and two-equation models. The new wall function approximates the analytical results of the Spalart-Allmaras turbulence model in the near wall region of the zero pressure gradient incompressible turbulent boundary layer. This approximate form can be applied as a unified eddy-viscosity-based wall function and a concise unified wall function of velocity distribution is proposed after integration. On this basis, the grid Reynolds number is introduced and a function for judging the position of the first layer grid is given. The new wall function is validated using the NACA0012 and the axisymmetric separated boundary layer. It is proved that the wall function is accurate and universal, and the grid requirements are lower in applications.

This study is concerned with numerical solutions of a Magneto-Thermo-Elasticity (MTE) model via a combination of energy-conserving finite difference method (FDM) with Richardson extrapolation methods (REMs). Firstly, by introducing two auxiliary functions and using second-order centered FDM and Crank-Nicolson method to approximate spatial and temporal derivatives, respectively, a two-level energy-conserving FDM is established for a MTE model. The priori estimation, solvability, and convergence are derived rigorously by using the discrete energy method. Secondly, to improve computational efficiency, a class of REMs are also designed by constructing the symbolic expansion of numerical solutions. Finally, numerical results confirm the efficiency of the proposed algorithms and the exactness of the theoretical findings.

In this paper, we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations. We first design an a posteriori error estimator and prove both the upper and lower bounds. Based on the a posteriori error estimator, we propose an adaptive planewave method. We then prove that the adaptive planewave approximations have the linear convergence rate and quasi-optimal complexity.

In this paper, we consider two stabilized second-order semi-implicit finite element methods for solving the Allen-Cahn and Cahn-Hilliard equations. Stabilized semi-implicit schemes are used for temporal discretization, and the finite element method is used for spatial discretization. It is shown that by adding a single linear term that is of the same order with the truncation error in time, the proposed methods are all unconditionally energy stable. Error estimates for the two schemes are also established. Numerical examples are presented to confirm the accuracy, efficiency and stability of the proposed methods.

In this paper, a lattice Boltzmann method is developed to simulate the water displacement of oil in porous media at the macroscopic-scale, which is built upon the generalized Darcy’s law while incorporating cross terms to consider the viscous coupling between two phases. The Buckley-Leverett equation is applied to describe the saturation front advance, thus allowing the determination of water saturation at shock front by using the Welge’s graphic method. We explore the effect of viscous coupling on the displacement by comparing the positions of the shock front under the conditions with different degrees of coupling and without viscous coupling. Results show that the advancing speed of shock front increases with the degree of coupling when the viscosity ratio of oil to water remains at 5. We also find that the effect of viscous coupling on the displacement can be neglected when the tolerance quantifying the degree of coupling equals the reciprocal of viscosity ratio. In addition, the effect of viscous coupling on the displacement is very sensitive to the change of viscosity ratio, which decreases monotonously with the decrease of viscosity ratio under the same coupling degree.

In this paper, we present an application of the isoparametric finite element for the Reissner-Mindlin plate problem on bounded domain with curved boundary. The discrete scheme is established by isoparametric quadratic triangular finite element combined with a numerical quadrature. Under the certain numerical quadrature, we prove the existence and uniqueness of the numerical solutions and the error estimates of optimal order in $H^1$-norm are given in details with the help of rigorous analysis. Finally, a numerical example is provided to verify the theoretical results.

In this paper, we propose a new way to construct the distribution function through the second-order polynomial approximation in terms of particle mass, momentum and energy. The new construction holds three distinguished features. First, the formulations are more concise as compared with the third-order truncated Hermite polynomial expansion which yields Grad’s 13-moment distribution function; Second, all moments of the present distribution function are determined from conservation laws; Third, these moments are closely linked to the most desirable variables, such as mass, momentum and energy. Then, this new distribution function is applied to construct a new gas kinetic flux solver. Numerical validations show that the proposed method recovers the Navier-Stokes solutions in the continuum regime. In addition, it outperforms Grad’s 13-moment distribution function in the transition regime, especially in the prediction of temperature and heat flux.

Aiming at modeling the cavitation bubble cluster, we propose a novel nonlinear dynamic cavitation model (NDCM) considering the second derivative term in Rayleigh-Plesset equation through strict mathematical derivation. There are two improvements of the new model: i) the empirical coefficients are eliminated by introduction of the nonuniform potential functions of $\psi_v$ and $\psi_c$ for growth and collapse processes respectively, and ii) only two model parameters are required, which both base on physical quantities – the Blake critical radius $R_b$ and the average maximum growth radius $R_m.$ The corresponding cavitation solver was developed by using OpenFOAM in which we implemented the modified momentum interpolation (MMI) method to ensure that the calculated results are independent of time step size. Three validation cases, namely numerical bubble cluster collapse, ultrasonic horn experiment, and hydrodynamic cavitation around slender body are employed. The results indicate that $\psi_v$ and $\psi_c$ can reveal the nonlinear characteristics for cavity accurately, and $R_b$ and $R_m$ can reflect the relevance between cavitation model and actual physical quantities. Moreover, it is discussed the potentiality of NDCM that is generally applied on the cavitating flow possessing with dispersed bubbly cloud.