In this paper, we present a Hessian recovery based linear finite element method to simulate the molecular beam epitaxy growth model with slope selection. For the time discretization, we apply a first-order convex splitting method and second-order Crank-Nicolson scheme. For the space discretization, we utilize the Hessian recovery operator to approximate second-order derivatives of a $C^0$ linear finite element function and hence the weak formulation of the fourth-order differential operator can be discretized in the linear finite element space. The energy-decay property of our proposed fully discrete schemes is rigorously proved. The robustness and the optimal-order convergence of the proposed algorithm are numerically verified. In a large spatial domain for a long period, we simulate coarsening dynamics, where 1/3-power-law is observed.

A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in $L^2$ and $H^1$ norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance $h$ does not appear explicitly in the parameter $β$ introduced by the Nitsche method. The theoretical results are confirmed numerically.

A computational code is developed for the numerical solution of one-dimensional transient gas-liquid flows using drift-flux models, in isothermal and also with phase change situations. For these two-phase models, classical upwind schemes such as Roe- and Godunov-type schemes are generally difficult to derive and expensive to use, since there are no treatable analytic expressions for the Jacobian matrix, eigenvalues and eigenvectors of the system of equations. On the other hand, the high-order compact finite difference scheme becomes an attractive alternative on these occasions, as it does not make use of any wave propagation information from the system of equations. The present paper extends the localized artificial diffusivity method for high-order compact finite difference schemes to solve two-phase flows with discontinuities. The numerical method has simple formulation, straightforward implementation, low computational cost and, most importantly, high-accuracy. The numerical methodology proposed is validated by solving several numerical examples given in the literature. The simulations are sixth-order accurate and it is shown that the proposed numerical method provides accurate approximations of shock waves and contact discontinuities. This is an essential property for simulations of realistic mass transport problems relevant to operations in the petroleum industry.

This paper presents a specific network architecture for approximation of the first Piola-Kirchhoff stress. The neural network enables us to construct the constitutive relation based on both macroscopic observations and atomistic simulation data. In contrast to traditional deep learning models, this architecture is intrinsic symmetric, guarantees the frame-indifference and material-symmetry of stress. Specifically, we build the approximation network inspired by the Cauchy-Born rule and virial stress formula. Several numerical results and theory analyses are presented to illustrate the learnability and effectiveness of our network.

The meandering river is an unstable system with the characteristic of nonlinearity, which results from the instability of the flow and boundary. Focusing on the hydrodynamic nonlinearity of the bend, we use the weakly nonlinear theory and perturbation method to construct the nonlinear evolution equations of the disturbance amplitude and disturbance phase of two-dimensional flow in meandering bend. The influence of the curvature, $Re$ and the disturbance wave number on the evolution of disturbance amplitude and disturbance phase are analyzed. Then, the spatial and temporal evolution of the disturbance vorticity is expounded. The research results show: that the curvature makes the flow more stable; that in the evolution of the disturbance amplitude effected by curvature, $Re$ and the disturbance wave number, exist nonlinear attenuation with damping disturbances, and nonlinear explosive growth with positive disturbances; that the asymmetry distribution of the disturbance velocities increases with the curvature; that the location of the disturbance vorticity’s core area changes periodically with disturbance phase, and the disturbance vorticity gradually attenuates/increases with the decrease of the disturbance phase in the evolution process of damping/positive disturbances. These results shed light on the construction of the interaction model of hydrodynamic nonlinearity and geometric nonlinearity of bed.

In the hybrid RANS-LES simulations, proper turbulent fluctuations should be added at the RANS-to-LES interface to drive the numerical solution restoring to a physically resolved turbulence as rapidly as possible. Such turbulence generation methods mostly need to know the distribution of the characteristic length scale of the background RANS model, which is important for the recovery process. The approximation of the length scale for the Spalart-Allmaras (S-A) model is not a trivial issue since the model’s one-equation nature. As a direct analogy, the approximations could be obtained from the definition of the Prandtl’s mixing length. Moreover, this paper proposes a new algebraic expression to approximate the intrinsic length scale of the S-A model. The underlying transportation mechanism of S-A model are largely exploited in the derivation of this new expression. The new proposed expression is employed in the generation of synthetic turbulence to perform the hybrid RANS-LES simulation of canonical wall-bounded turbulent flows. The comparisons demonstrated the feasibility and improved performance of the new length scale on generating synthetic turbulence at the LES inlet.

In this paper, we present a linearized compact difference scheme for one-dimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions. The initial singularity of the solution is considered, which often generates a singular source and increases the difficulty of numerically solving the equation. The Crank-Nicolson technique, combined with the midpoint formula and the second-order convolution quadrature formula, is used for the time discretization. To increase the spatial accuracy, a fourth-order compact difference approximation, which is constructed by two compact difference operators, is adopted for spatial discretization. Then, the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space. Finally, numerical experiments are given to support our theoretical results.

In this article, a new characteristic finite difference method is developed for solving miscible displacement problem in porous media. The new method combines the characteristic technique with mass-preserving interpolation, not only keeps mass balance but also is of second-order accuracy both in time and space. Numerical results are presented to confirm the convergence and the accuracy in time and space.

In this paper, we present a new vertex-centered arbitrary Lagrangian-Eulerian (ALE) finite volume scheme for two-dimensional compressible flow. In our scheme, the momentum equation is discretized on the vertex control volume, while the mass equation and the energy equation are discretized on the sub-cells which are included in the vertex control volume. We attain the average of the fluid velocity on the vertex control volume directly by solving the conservation equations. Then we can obtain the fluid velocity at vertex with the reconstructed polynomial of the velocity. This fluid velocity is chosen as the mesh velocity, which makes the mesh move in a Lagrangian manner. Two WENO (Weighted Essentially Non-Oscillatory) reconstructions for the density (the total energy) and the velocity are used to make our scheme achieve the anticipated accuracy. Compared with the general vertex-centered schemes, our scheme with the new approach for the space discretization can simulate some multi-material flows which do not involve large deformations. In addition, our scheme has good robustness, and some numerical examples are presented to demonstrate the anticipated accuracy and the good properties of our scheme.

In this paper, two fully-discrete local discontinuous Galerkin (LDG) methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria. The numerical methods are linear and decoupled, which greatly improve the computational efficiency. In order to resolve the time level mismatch of the discretization process, a special time marching method with high-order accuracy is constructed. Under the condition of slight time step constraints, the optimal error estimates of this method are given. Moreover, the theoretical results are verified by numerical experiments. Real simulations show the patterns of spots, rings, stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.

This study addresses the parameter identification problem in a system of time-dependent quasi-linear partial differential equations (PDEs). Using the integral equation method, we prove the uniqueness of the inverse problem in nonlinear PDEs. Moreover, using the method of successive approximations, we develop a novel iterative algorithm to estimate sorption isotherms. The stability results of the algorithm are proven under both a priori and a posteriori stopping rules. A numerical example is given to show the efficiency and robustness of the proposed new approach.

In this paper, a new cracked stiffener model for the stiffener with a part-through and open crack is proposed, considering the compatibility condition of displacements between the plate and the stiffener. Based on the first-order shear deformation theory, the free vibration of stiffened isotropic plates with cracked stiffeners are investigated for the first time. The description of the crack parameters is based on the continuous equivalent bending stiffness and equivalent depth of the cracked beam, and it takes into consideration of shear deformation, bending-extensional coupling vibration, and eccentricity between the stiffeners and the plate. The stiffened plates with single or multiple cracked stiffeners are formulated and discussed. The Ritz method with the modified characteristic functions is applied to demonstrate the effects of crack parameters (crack depth and location) coupling with the position and number of the cracked stiffeners on the vibration frequencies and modes of the stiffened plate. The validity and accuracy of the present solutions are verified through convergence studies and compared with the finite element results.