Efficient, divergence-free, high-order MHD on 3D spherical meshes with optimal geodesic meshing

ABSTRACT There is a great need in several areas of astrophysics and space physics to carry out high order of accuracy, divergence-free MHD simulations on spherical meshes. This requires us to pay careful attention to the interplay between mesh quality and numerical algorithms. Methods have been designed that fundamentally integrate high-order isoparametric mappings with the other high accuracy algorithms that are needed for divergence-free MHD simulations on geodesic meshes. The goal of this paper is to document such algorithms that are implemented in the geodesic mesh version of the RIEMANN code. The fluid variables are reconstructed using a special kind of WENO-AO algorithm that integrates the mesh geometry into the reconstruction process from the ground-up. A novel divergence-free reconstruction strategy for the magnetic field that performs efficiently at all orders, even on isoparametrically mapped meshes, is then presented. The MHD equations are evolved in space and time using a novel ADER predictor algorithm that is efficiently adapted to the isoparametrically mapped geometry. The application of one-dimensional and multidimensional Riemann solvers at suitable locations on the mesh then provides the corrector step. The corrector step for the magnetic field uses a Yee-type staggering of magnetic fields. This results in a scheme with divergence-free update for the magnetic field. The use of ADER enables a one-step update that only requires one messaging operation per complete timestep. This is very beneficial for parallel processing. Several accuracy tests are presented as are stringent test problems. PetaScale performance is also demonstrated on the largest available supercomputers.

A Predictor–Corrector Algorithm for Monotone Linear Complementarity Problems in a Wide Neighborhood

We propose a new primal-dual interior-point predictor–corrector algorithm in Ai and Zhang’s wide neighborhood for solving monotone linear complementarity problems (LCP). Based on the understanding of this neighborhood, we use two new directions in the predictor step and in the corrector step, respectively. Especially, the use of new corrector direction also reduces the duality gap in the corrector step, which has good effects on the algorithm’s convergence. We prove that the new algorithm has a polynomial complexity of [Formula: see text], which is the best complexity result so far. In the paper, we also prove a key result for searching for the best step size along some direction. Considering local convergence, we revise the algorithm to be a variant, which enjoys both complexity of [Formula: see text] and Q-quadratical convergence. Finally, numerical result shows the effectiveness and superiority of the two new algorithms for monotone LCPs.

A corrector–predictor path-following algorithm for semidefinite optimization

A corrector–predictor algorithm is proposed for solving semidefinite optimization problems. In each two steps, the algorithm uses the Nesterov–Todd directions. The algorithm produces a sequence of iterates in a neighborhood of the central path based on a new proximity measure. The predictor step uses line search schemes requiring the reduction of the duality gap, while the corrector step is used to restore the iterates to the neighborhood of the central path. Finally, the algorithm has [Formula: see text] iteration complexity.

A predictor-corrector path-following algorithm for symmetric optimization based on Darvay's technique

In this paper, we present a predictor-corrector path-following interior-point algorithm for symmetric cone optimization based on Darvay's technique. Each iteration of the algorithm contains a predictor step and a corrector step based on a modification of the Nesterov and Todd directions. Moreover, we show that the algorithm is well defined and that the obtained iteration bound is o(?rlogr?/?), where r is the rank of Euclidean Jordan algebra.

THE GROWING STRING METHOD FOR FLOWS OF NEWTON TRAJECTORIES BY A SECOND-ORDER METHOD

The reaction path is an important concept of theoretical chemistry. We use a definition with a reduced gradient (see Quapp et al., Theor Chem Acc100:285, 1998), also named Newton trajectory (NT). To follow a reaction path, we design a numerical scheme for a method for finding a transition state between reactant and product on the potential energy surface: the growing string (GS) method. We extend the method (see W. Quapp, J Chem Phys122:174106, 2005) by a second-order scheme for the corrector step, which includes the use of the Hessian matrix. A dramatic performance enhancement for the exactness to follow the NTs, and a dramatic reduction of the number of corrector steps are to report. Hence, we can calculate flows of NTs. The method works in nonredundant internal coordinates. The corresponding metric to work with is curvilinear. The GS calculation is interfaced with the GamessUS package (we have provided this algorithm on ). Examples for applications are the HCN isomerization pathway and NTs for the isomerization C7ax ↔ C5 of alanine dipeptide.