High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there is still room for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a fifth-order accuracy for the point-wise values at a cell interface due to the use of standard WENO reconstruction, and the slopes of the initial non-equilibrium states have to be reconstructed from the cell interface values and cell averages again. The same order of accuracy for slopes as the original WENO scheme cannot be achieved. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) [2] method is implemented for reconstruction. Equipped with WENO-AO reconstruction, the performance enhancement of HGKS is fully explored. WENO-AO not only provides the interface values, but also the slopes. In other words, a whole polynomial inside each cell is provided by the WENO-AO reconstruction. The available polynomial may not benefit to the high-order schemes based on the Riemann solver, where only points-wise values at the cell interface are needed. But, it can be fully utilized in the HGKS. As a result, the HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in the previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction. In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases.

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.

In this work, we show that the modified ghost fluid method might suffer pressure mismatch at material interfaces and thus leads to inaccurate numerical results when directly applied to long term simulations of multi-medium flow problems with an axisymmetric source term. We disclose the underlying reason and then develop a technique of linear distribution to take into account the effect of the axisymmetric source on the definition of ghost fluid states. In order to faithfully consider the effect of the source term, the interfacial conditions related to derivatives are derived and linear distributions of ghost fluid states are constructed based on a generalized axisymmetric multi-medium Riemann problem. Theoretical analysis and numerical results show that the modified ghost fluid method with axisymmetric source correction (MGFM/ASC) can effectively eliminate the pressure error.

In this work, the full vectorial beam propagation methods (BPMs) are adopted for the calculations of the electromagnetic wave propagation from the two dimensional and three dimensional optical waveguide structures. First, the full vectorial BPM for the three dimensional optical waveguide structures is introduced. Next, in the transverse directions of the considered waveguide structures, we adopt the second order finite difference method to discretize the electromagnetic components. Then, the Lanczos/Arnoldi fast solvers are adopted to find the leading eigenvalues and eigenvectors of the square root operator in the BPM process of the optical waveguide structures. Furthermore, we propose the rational [($p$−1)/$p$] Padé approximation to approximate the exponential operator in the BPM process. To demonstrate the efficiency of the numerical solvers, the two dimensional symmetric and unsymmetric problems are considered, and good convergence results are obtained. Furthermore, the resulting full-vectorial BPM is adopted to simulate the wave propagation among the three dimensional rib and taper waveguide structures. Numerical results demonstrate the efficiency of the proposed method with respect to both the accuracies and convergence results.

Chlorophyll in leaves is tightly associated with physiological status of plants. Chemical extraction or hyperspectral estimation is the conventional method to estimate the concentration of Chlorophyll in leaves. However, chemical extraction is invasive and time consuming, and hyperspectral method is extremely sensitive to background light. In this paper, we develop a quantitative photoacoustic imaging technique based on a finite-element-based reconstruction algorithm accelerated by a multicore GPU card to image morphological features and derive distribution of Chlorophyll A in rice leaves. The results suggest that this new method holds great potential in various studies of plant physiology.

This paper is concerned with the classical problem of wave propagation in discrete models of nonuniform resolution. We extend the traditional asynchronous variational integrators (AVIs) method to higher order and couple different spatial elements to adapt to nonuniform meshes. We show that the extension of AVIs method is stable, convergent and may reduce the spurious inter-grid reflection across meshes with different sizes. Numerical experiments are provided to verify the stability and convergence of the extended AVIs. The total energy is numerically conserved in our experiments.

Conventional shot-record reverse time migration (RTM) suffers from a high computational burden when dealing with massive data. The computational cost of RTM can be reduced by shot-encoding techniques, and plane-wave encoding is a commonly used and effective shot-encoding scheme. However, plane-wave encoding requires long time padding to avoid information loss, which decreases the efficiency of the time-domain wavefield extrapolator, and the time padding becomes longer with the increasing distance between multiple sources. The piecewise plane-wave encoding scheme cuts multiple sources into several segments prior to implementing plane-wave encoding, hence reduces the time padding, but brings new crosstalk due to the mutual interference between shots in different source segments. We suppress the crosstalk artifacts by a new bipolar-bisection amplitude encoding method, in which half of the encoding array of each migration is different from that of any other migrations to reduce the number of crosstalk terms with as few migrations as possible. We embed the bipolar-bisection method into piecewise plane-wave encoding. Compared with plane-wave encoding, the proposed scheme requires considerably shorter time padding and thus works more efficiently and can generate a qualified imaging result. The feasibility of the proposed method is tested on the 2D SEG/EAGE salt model and the Marmousi model.

In this paper, a new approach is proposed to analyse the stability of high-order staggered-grid finite difference schemes for the three-dimensional (3D) poroelastic wave propagation. The standard staggered-grid schemes with different order accuracy in space are constructed based on the first-order hyperbolic velocity-stress system of the governing equations (i.e., Biot's equations). The new analysis method is based on von Neumann analysis. The obtained 3D stability is an explicit restriction for time step, which only depends on the coefficients of the difference operators and the material parameters of poroelastic media and so it can be computed easily. Moreover, the analysis has good generality and can be applied directly to the staggered-grid schemes for 3D elastic wave. Numerical computations with the perfectly matched layer in split formation are implemented to illustrate the effectiveness of the schemes for 3D poroelastic wave propagation. The method in this paper can be expected to analyse the stability for other staggered-grid schemes.

In a general polygonal domain, possibly nonconvex and multi-connected (with holes), the time-dependent Ginzburg–Landau equation is reformulated into a new system of equations. The magnetic field $B$:=∇×A is introduced as an unknown solution in the new system, while the magnetic potential A is solved implicitly through its Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method is proposed for solving the new system. The convergence of numerical solutions is proved based on a compactness argument by utilizing the maximal $L^p$-regularity of the discretized equations. Compared with the Hodge decomposition method proposed in [27]_, the new method has the advantage of approximating the magnetic field B directly and converging for initial conditions that are incompatible with the external magnetic field. Several numerical examples are provided to illustrate the efficiency of the proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new method is superior to the method in [27] for approximating the magnetic field.

We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].

This is the fourth installment in our series on implementing the discontinuous Galerkin (DG) method as an open source MATLAB / GNU Octave toolbox. Similarly to its predecessors, this part presents new features for application developers employing DG methods and follows our strategy of relying on fully vectorized constructs and supplying a comprehensive documentation. The specific focus of the current work is the newly added generic problem implementation framework and the highly customizable model-coupling interface for multi-domain and multi-physics simulation tools based on this framework. The functionality of the coupling interface in the FESTUNG toolbox is illustrated using a two-way coupled free-surface / groundwater flow system as an example application.