High-order Stencil Research Articles

Universally applicable uniaxial perfect matched layer formulation for explicit and implicit finite difference time domain algorithms

An important advance in the finite difference time domain (FDTD) algorithms is Berenger's perfect matched layer (PML), along with its derivatives. In recent years several new explicit and implicit FDTD algorithms have appeared on the scene, of which each requires a specific PML. To simplify programming, the authors derive a universal uniaxial PML formulation, which is universally applicable to the Yee FDTD, FDTD with higher-order stencils and implicit FDTD algorithms. Numerical results show the efficiency and versatility of the new method.

A Constrained‐Transport Magnetohydrodynamics Algorithm with Near‐Spectral Resolution

Numerical simulations including magnetic fields have become important in many fields of astrophysics. Evolution of magnetic fields by the constrained-transport algorithm preserves magnetic divergence to machine precision and thus represents one preferred method for the inclusion of magnetic fields in simulations. We show that constrained transport can be implemented with volume-centered fields and hyperresistivity on a high-order finite-difference stencil. In addition, the finite-difference coefficients can be tuned to enhance high-wavenumber resolution. Similar techniques can be used for the interpolations required for dealiasing corrections at high wavenumber. Together, these measures yield an algorithm with a wavenumber resolution that approaches the theoretical maximum achieved by spectral algorithms. Because this algorithm uses finite differences instead of fast Fourier transforms, it runs faster and is not restricted to periodic boundary conditions. In addition, since the finite differences are spatially local, this algorithm is easily scalable to thousands of processors. We demonstrate that, for low-Mach number turbulence, the results agree well with a high-order, non-constrained-transport scheme with Poisson divergence cleaning.

A posteriori error estimation by postprocessor independent of method of flowfield calculation

We consider a postprocessor that is able to analyze the flow-field generated by an external (unknown) code so as to determine the error of useful functionals. The residuals engendered by the action of a high-order finite-difference stencil on a numerically computed flow-field are used for adjoint based a posteriori error estimation. The method requires information on the physical model (PDE system), flowfield parameters and corresponding grid and may be constructed without availability of detailed information on the numerical method used for the flow computation.