In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.

This paper focuses on discussing Newton’s method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the mixed element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

Fluid flows in microfluidic devices are often characterized by non-Newtonian rheology with non-linear wall slip behavior also observed. This work solves this problem class with the lattice Boltzmann method (LBM), proposing new advanced boundary scheme formulations to model the joint contribution of non-linear rheology and non-linear wall slip laws in application to microchannels of planar and circular cross-section. The non-linear stress-strain-rate relationship of the microflow is described by a generalized Newtonian model where the viscosity function follows the Sisko model. To guarantee that LBM steady-state solutions are not contaminated by numerical errors that depend on the viscosity local value, the two-relaxation-time (TRT) collision is adopted. The fluid-wall accommodation model considers different slip laws, such as the Navier linear, Navier non-linear, empirical asymptotic and Hatzikiriakos slip laws. They are transcribed into the LBM framework by adapting the local second-order boundary (LSOB) scheme strategy to this problem class. Theoretical and numerical analyses developed for a steady and slow viscous fluid within 2D slit and 3D circular pipe channels demonstrate the parabolic level of accuracy of the developed LSOB scheme throughout the considered non-linear slip and non-Newtonian models.

In many applications, such as plasma edge simulation of a nuclear fusion reactor, a coupled PDE/kinetic description is required. Such systems can be solved with a coupled finite-volume/Monte-Carlo method. Different procedures have been proposed to estimate the source terms in the finite volume part that appear from the Monte Carlo part of the simulation. In this series of papers, we present a systematic (analytical and numerical) comparison of the variance and computational cost of a coherent set of such estimation procedures. The comparison is based on an invariant imbedding procedure, in which systems of ordinary differential equations (ODEs) are derived that quantify the statistical error and computational cost of each estimator. In this paper, we discuss analog and collision simulation and estimation procedures. We analyze in detail a scenario with forward-backward scattering in a one-dimensional slab, uncovering and quantifying the effects determining the performance of the estimation procedures.

We propose a kind of second-order stabilized Crank-Nicolson scheme which can be applied to three types of Cahn-Hilliard model with dynamic boundary conditions. We give the corresponding proof of stability and convergence theoretically which takes the reaction rate dependent dynamic boundary conditions as an example. We verify the effectiveness and universality of our proposed scheme by conducting some typical numerical simulations and comparing with the literature works. It’s found that second-order scheme takes much less CPU time than the first-order scheme to reach the same final time.

The re-splitting $δf$ method (Ye et. al CPC 2020) has been implemented into the global semi-Lagrangian gyrokinetic code NLT (Lei Ye et al. 2016, JCP) for the simulations of electromagnetic micro-turbulence in tokamak plasmas. The re-splitting method can be incorporated with the numerical Lie transform method, which is an essential numerical scheme for NLT code, to mitigate the cancellation problem appears in the gyrokinetic $p_{||}$-formulation with gyrokinetic ions and drift-kinetic electrons. With this method, the ion temperature gradient mode (ITG), the kinetic ballooning mode (KBM) and trapped electron mode (TEM) are simulated by NLT and the results are well benchmarked with other gyrokinetic simulation codes.

Generally, to apply the MUltiple SIgnal Classification (MUSIC) algorithm for the rapid imaging of small objects, complete elements of the multi-static response (MSR) matrix must be collected. However, in some real-world applications in microwave imaging, diagonal elements of the MSR matrix are unknown. Nevertheless, it is possible to obtain imaging results using a traditional approach but theoretical reason of the applicability has not been investigated yet. In this paper, we consider the application of MUSIC for a fast identification of small objects from collected MSR matrix in both transverse magnetic (TM) and transverse electric (TE) polarizations. In order to examine the applicability, fundamental limitation, and various properties of MUSIC, we establish mathematical structure of the three imaging functions and explore that the main factors of the imaging functions are Bessel function of order zero, one, and two. The established structures demonstrate why the existence and location of small objects can be retrieved via MUSIC without the diagonal elements of the MSR matrix. Results of numerical simulations with noise-corrupted synthetic data are also provided to support the identified structures.

In this paper, the model reduction method based on $k$-nearest-neighbors is provided for the parametrized nonlocal partial differential equations (PDEs). In comparison to standard local PDEs, the stiffness matrix of the corresponding nonlocal model loses sparsity due to the nonlocal interaction parameter $δ.$ Specially the nonlocal model contains uncertain parameters, enhancing the complexity of computation. In order to improve the computation efficiency, we combine the $k$-nearest-neighbors with the model reduction method to construct the efficient surrogate models of the parametrized nonlocal problems. This method is an offline-online mechanism. In the offline phase, we develop the full-order model by using the quadratic finite element method (FEM) to generate snapshots and employ the model reduction method to process the snapshots and extract their key characters. In the online phase, we utilize $k$-nearest-neighbors regression to construct the surrogate model. In the numerical experiments, we first verify the convergence rate when applying quadratic FEM to the nonlocal problems. Subsequently, for the linear and nonlinear nonlocal problems with random inputs, the numerical results illustrate the efficiency and accuracy of the surrogate models.

In recent decades, the gas-kinetic unified algorithm (GKUA) is proposed to simulate the gas flow in the whole flow regimes. In previous studies, GKUA was extended to problems of flows around the aircraft and internal flows considering the influence of rotational energy, as well as the flows around the aircraft considering the influence of vibrational energy. In this paper, we try to apply the model considering the effect of vibrational energy to the engine internal and external mixed flow problem, and build a numerical simulation framework of engine internal and external mixed flow considering the effect of vibrational energy. The reliability of the algorithm is verified by using one-dimensional shock tube problem and two-dimensional nozzle internal flow problem. The results show that the model can describe the flow problem of high temperature nozzle internal flow. Then the two-dimensional nozzle internal and external mixed flow problem of 25N attitude control engine is studied, the unsteady evolution and non-equilibrium effect of the mixed flow field are analyzed, and the flow parameter distribution of the mixed flow field without internal energy excitation, considering rotational energy excitation and considering vibrational energy excitation are compared.