The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22-44 and Zhang S., Jiang S and Shu C.-W. (2008), J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273-305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216-238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

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.

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws, we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in earlier work, we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

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.

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.