In this paper, we propose a class of high order locally divergence-free spectral-discontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r,ϕ,z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r,z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the locally divergence-free set for the magnetic filed, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

We present a CUDA accelerated implementation of the Characteristic/Scan Conversion algorithm to generate narrow band signed distance fields in logically Cartesian grids. We outline an approach of task and data management on GPUs based on an input of a closed triangulated surface with the aim of reducing pre-processing and mesh-generation times. The work demonstrates a fast signed distance field generation of triangulated surfaces with tens of thousands to several million features in high resolution domains. We present improvements to the robustness of the original algorithm and an overview of handling geometric data.

We develop a numerical model that reproduces the thermal equilibrium and the spin transfer mechanisms in magnetic materials. We analyze the coherent two-particle spin exchange interaction and the electron-electron collisions. Our study is based on a quantum atomistic approach and the particle dynamics is performed by using a Monte Carlo technique. The coherent quantum evolution of the atoms is interrupted by instantaneous collisions with itinerant electrons. The collision processes are associated to the quantum collapse of the local atomic wave function. We show that particle-particle interactions beyond the molecular field approximation can be included in this framework.
Our model is able to reproduce the thermal equilibrium and strongly out-of-equilibrium phenomena such as the ultrafast dynamics of the magnetization in nanomaterials.

In this work, we have proven the inhomogeneous obstacle can be uniquely determined by the measured acoustic pressure data and have proposed an extended multilevel sampling method. The extended recovery approach only applies the matrix-vector operations to estimate the inhomogeneous media from the received partial data. In practice, the method is capable of reconstructing the objects of different shapes and locations, robust against noise, computationally fairly cheap and easy to carry out. We can consider it as a simple and direct algorithm to supply satisfactory initial locations for the application of any existing more refined and precise but computationally more demanding techniques to achieve accurate reconstructions of physical features of scatterers.

This paper presents an implementation of the adaptive hybrid WENO (weighted essentially non-oscillatory) scheme based on our previous investigations for compressible multi-medium flows (Liu and Hu, J. Comput. Phys., 342 (2017), 43-65). In this study a simple and efficient method is developed for Euler equations and Navier-Stokes equations arising from the conservation laws. A class of high order weighted essentially non-oscillatory (WENO) schemes are applied to resolve the complicated flow structures and shock waves. Classical WENO schemes are computationally expensive in calculating the non-linear weight and smoothness indicators. We propose a block-structured adaptive mesh method together with a modified hybrid-WENO scheme to reduce the cost, the reconstruction is only performed at non-smooth region. Comparisons of WENO scheme with various smoothness indicators and different Lax-Friedrich flux vector splitting methods are performed on block structured adaptive mesh. Benchmark tests show present adaptive hybrid WENO method is low-dissipative and highly robust. The 2-D/3-D shock wave boundary layer interaction are simulated to verify the efficiency of present AMR (adaptive mesh refinement) solver in predicting turbulent flow

As the computing power of the latest parallel computer systems increases dramatically, the fully coupled space-time solution algorithms for the time-dependent system of PDEs obtain their popularity recently, especially for the case of using a large number of computing cores. In this space-time algorithm, we solve the resulting large, space, nonlinear systems in an all-at-once manner and a robust and efficient nonlinear solver plays an essential role as a key kernel of the whole solution algorithm. In the paper, we introduce and study some parallel nonlinear space-time preconditioned Newton algorithm for the space-time formulation of the thermal convective flows at high Rayleigh numbers. In particular, we apply an adaptive nonlinear space-time elimination preconditioning technique to enhance the robustness of the inexact Newton method, in the sense that an inexact Newton method can converge in a broad range of physical parameters in the multi-physical heat fluid model. In addition, at each Newton iteration, we find an appropriate search direction by using a space-time overlapping Schwarz domain decomposition algorithm for solving the Jacobian system efficiently. Some numerical results show that the proposed method is more robust and efficient than the commonly-used Newton-Krylov-Schwarz method.

The modified dimension-by-dimension finite volume (FV) WENO method on Cartesian grids proposed by Buchmüller and Helzel can retain the full order of accuracy of the one-dimensional WENO reconstruction and requires only one flux computation per interface. The high-order accurate conversion between face-averaged values and face-center point values is the main ingredient of this method. In this paper, we derive sixth-order accurate conversion formulas on three-dimensional Cartesian grids. It is shown that the resulting modified FV WENO method is efficient and high-order accurate when applied to smooth nonlinear multidimensional problems, and is robust for calculating non-smooth nonlinear problems with strong shocks.

Ill-posed Cauchy problems for elliptic partial differential equations appear in many engineering fields. In this paper, we focus on stable reconstruction methods for this kind of inverse problems. Using kernels that reproduce Hilbert spaces $H^m$(Ω), numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints (LSQI). A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm using generalized singular value decomposition of matrices and Lagrange multipliers is proposed to solve the LSQI problem. Numerical experiments of two-dimensional cases verify our proved convergence results. By comparing with solutions of MFS and FEM under the discrete Tikhonov regularization by RKHS under same Cauchy data, we demonstrate that our method can reconstruct stable and high accuracy solutions for noisy Cauchy data.

Consider an inverse obstacle scattering problem in an open space which is filled with a homogeneous and isotropic elastic medium. The inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, a new approach is proposed for numerical solution of the inverse problem. By introducing two scalar potential functions, the method uses the Helmholtz decomposition to split the displacement of the elastic wave equation into the compressional and shear waves, which satisfy a coupled boundary value problem of the Helmholtz equations. The domain derivative is studied for the coupled Helmholtz system. In particular, we show that the domain derivative of the potentials is the Helmholtz decomposition of the domain derivative of the displacement for the elastic wave equation. Numerical results are presented to demonstrate the effectiveness of the proposed method.

This paper is concerned with unconditionally optimal error estimate of the linearized Galerkin finite element method for solving the two-dimensional and three-dimensional Kuramoto-Tsuzuki equations, while the classical analysis for these nonlinear problems always requires certain time-step restrictions dependent on the spatial mesh size. The key to our analysis is to obtain the boundedness of the numerical approximation in the maximum norm, by using error estimates in certain norms in the different time level, the corresponding Sobolev embedding theorem, and the inverse inequality. Numerical examples in both 2D and 3D nonlinear problems are given to confirm our theoretical results.

In this paper, an adjoint-based h-adaptive high-order reconstructed DG (rDG) method is introduced for solving the two dimensional steady-state compressible Euler equations. Based on the hybrid reconstruction strategy developed in [9, 28], adjoint-based a posteriori error estimation is further derived and developed for h-adaption. The formulation of error indicator is carefully investigated in order to deliver better approximation with respect to the error in the computed output functional. In order to evaluate the performance of the proposed method, an adjoint-based h-adaptive rDG(p₁ p₂) method is implemented, in which a hybrid p₁ p₂ reconstruction and a hybrid p₂ p₃ reconstruction are adopted in the primal solver and the adjoint solver to obtain the primal solution and the adjoint solution, respectively. A number of typical test cases are selected to assess the performance of the adjoint-based h-adaptive hybrid rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques based on adjoint-based error estimate presented in this work demonstrates its capacity in reducing the error with respect to the computed output functional and improving the level of accuracy for numerical simulations of the compressible inviscid flows.

It is well-established that the accurate simulation of hypersonic rarefied gas flows cannot be accomplished by Navier-Stokes-Fourier (NSF) equations since they are applicable only for small deviations from local thermodynamic equilibrium. To accurately and efficiently solve this problem, a nonlinear-coupled-constitutive-relation (NCCR) model was developed from Eu's generalized hydrodynamic equations, which are consistent with the laws of irreversible thermodynamics. In this paper, Myong's decomposed solver is extended for solving three-dimensional diatomic nonlinear coupled algebraic constitutive equations. Subsequently, a reliable undecomposed hybrid algorithm is proposed for the complete solution of NCCR model through combining the merits of fixed-point and Newton's iterations. The new-developed computational method is validated by high-speed rarefied flows around a cylinder and the Apollo command module. Computation shows that the undecomposed hybrid algorithm makes a great improvement in computational robustness and accuracy. Moreover, this nonlinear constitutive model yields solutions in better agreement with DSMC than NSF model. The results indicate that NCCR model is capable of capturing the physical flow properties away from equilibrium and demonstrate its great potential capability in the further application.

A test particle module is developed under BOUT++ framework. The guiding-center orbits in real divertor geometry can be calculated. The turbulence transport is implemented by random walk model. Impurity migration patterns under different turbulence transport levels are simulated by this module. As transport increase, migration pattern is modified significantly. More particles are lost and the lost at low field side boundary increase significantly.

We study the effect of the anneal path control per qubit, a new user control feature offered on the D-Wave 2000Q quantum annealer, on the performance of quantum annealing for solving optimization problems by numerically solving the time-dependent Schrödinger equation for the time-dependent Hamiltonian modeling the annealing problems. The anneal path control is thereby modeled as a modified linear annealing scheme, resulting in an advanced and retarded scheme. The considered optimization problems are 2-SAT problems with 12 Boolean variables, a known unique ground state and a highly degenerate first excited state. We show that adjustment of the anneal path control can result in a widening of the minimal spectral gap by one or two orders of magnitude and an enhancement of the success probability of finding the solution of the optimization problem. We scrutinize various iterative methods based on the spin floppiness, the average spin value, and on the average energy and describe their performance in boosting the quantum annealing process.