Based on the double shockwave approximation procedure, a local Riemann solver for strongly nonlinear equations of state (EOS) such as the Jones-Wilkins-Lee (JWL) EOS is presented, which has suppressed successfully numerical oscillation caused by high-density ratio and high-pressure ratio across the interface between explosion products and air. The real ghost fluid method (RGFM) and the level set method have been used for converting multi-medium flows into pure flows and for implicitly tracking the interface, respectively. A fifth order finite difference weighted essentially non-oscillatory (WENO) scheme and a third order TVD Runge-Kutta method are utilized for the spatial discretization and the time advance, respectively. An enclosed-type MPI-based parallel methodology for the RGFM procedure on a uniform structured mesh is presented to realize the parallelization of three-dimensional {\color{red}(3D)} air explosion. The overall process of 3D air explosion of both TNT and aluminized explosives has been successfully simulated.

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three dimensional photonic crystals with face centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of $5.184$ million dimension eigenvalue problems within $50$ to $104$ minutes on a workstation with two Intel Quad-C ore Xeon X5687 3.6 GHz CPUs.

In this work, we study the unsteady axisymmetric mixed convection boundary layer flow and heat transfer of non-Newtonian power law fluid over a cylinder. Different from most classical works, the temperature dependent variable fluid viscosity and thermal conductivity are taken into account in highly coupled velocity and temperature fields. The motion of the fluid can be modeled by a time-dependent nonlinear parabolic system in cylindrical coordinates, which is solved numerically by using Chebyshev spectral method along with the strong stability preserving (SSP) third order Runge-Kutta time discretization. We apply the numerical solver to problems with different power law indices, viscosity parameter, thermal conductivity parameter and Richardson numbers, and compute up to the steady state. The numerical solver is checked by testing the spectral convergence of the numerical approximation to a smooth exact solution of the PDEs with source terms. Moreover, the combined effects of pertinent physical parameters on the flow and heat transfer characteristics are analyzed in detail.

In this article, we consider the time-dependent Maxwell’s equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L ∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue.

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.