The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM uses the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high quality unstructured meshes due to the complexity of fracture's geometrical morphology. In this paper, we clarify that the DFM actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor to describe fractures as one-dimensional line Dirac delta functions contained in permeability tensor. A finite element DFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that our DFM automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes are demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.

The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes. The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper.

This work is devoted to the numerical computation of complex band structure $\k=\k(\omega)\in\mathbb C^3$ for positive $\omega$ of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in $\k\in\mathbb C^3$ and to view the remaining one as the eigenvalue of a quadratic operator pencil derived from Maxwell's equations. Then Yee's scheme is employed to discretize $\nabla\times$ and $\k\times$ operators in this quadratic operator pencil. Distinct from the others' works which either ignore or directly exploit the Hamiltonian structure of the spectrum of the resulting QEP, we reformulate this QEP into an equivalent $\top$-palindromic QEP to facilitate the use of superior structure-preserving algorithms. Ultimately we rely on the structured Arnoldi algorithm, namely the G$\top$SHIRA algorithm, to compute eigenvalues of a $\top$-skew-Hamiltonian pair which are near or in $[-2,2]$, a much narrower region than the whole positive real axis in the origin problem. Moreover, to accelerate the inner iterations of the G$\top$SHIRA algorithm, we propose the preconditioning technique, making most of the eigenmatrix, which can essentially be seen as the Kronecker product of three discrete Fourier transformation matrices, of the commutative discretized $\partial_x,\partial_y,\partial_z$ operators. The advantage of our method is discussed in detail and corroborated by several numerical results.

We present numerical schemes for the incompressible Navier–Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematic equations. The result is a class of extremely efficient Navier–Stokes solvers. Full time accuracy is achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we prove unconditional stability of the new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier–Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C0 elements to implement.

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