This paper is devoted to developing the Il'in-Allen-Southwell (IAS) parameter-uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection-diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct the IAS scheme for 1-D systems and then extend the scheme to 2-D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme is formally second order accurate in the sense that it is second order convergent when the perturbation parameter $\varepsilon$ is not too small and when the $\varepsilon$ is sufficiently small, the scheme is first order convergent in the discrete maximum norm uniformly in $\varepsilon$.

In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of $H^1$-conforming finite elements, namely the $CP_2$-$P_1$ elements, for electric field and multiplier is studied and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed mixed $H^1$-conforming finite element method.

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In earlier work, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

In earlier work, two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations. The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.