Yaman Güçlü

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

Abstract The Convected Scheme (CS) is a ‘forward-trajectory’ semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so-called ‘moving cells’ along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently (Guclu and Hitchon, 2012 [1] ) a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems. Here the CS is applied to the solution of the Vlasov–Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D–1V test cases, both in the linear and the non-linear regimes.

Finite difference weighted essentially non-oscillatory schemes with constrained transport for 2D ideal Magnetohydrodynamics

Summary form only given. A novel algorithm based on a high order finite difference adaptive WENO scheme [2] will be presented to solve the 2D ideal Magnetohydrodynamics (MHD) equations. The algorithm will use the unstaggered Constrained Transport technique from the magnetic potential advection constrained transport method [2] to satisfy the divergence-free constraint of the magnetic field. However the treatment of our algorithm is significantly different from [2]. The new feature will be (1) the method is finite difference type, (2) high order in both time (4th-order) and space (3rd-order), (3) all the conservative quantities are essentially non-oscillatory, (4) Adaptive Mesh Refinement will be used as the base framework to increase resolution. Convergence study will be done on the smooth problem. 2D/2.5D benchmark problems such as rotated Brio-Wu shock tube, Orszag-Tang and Cloud-shock interaction will be presented. We expect our algorithm is robust, essentially non-oscillatory and can capture shock waves and discontinuities well.

The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes

High-order temporal discretizations for hyperbolic conservation laws have historically been formulated as either a method of lines (MOL) or a Lax--Wendroff method. In the MOL viewpoint, the partial differential equation is treated as a large system of ordinary differential equations (ODEs), where an ODE tailored time integrator is applied. In contrast, Lax--Wendroff discretizations immediately convert Taylor series in time to discrete spatial derivatives. In this work, we propose the Picard integral formulation (PIF), which is based on the method of modified fluxes, and is used to derive new Taylor and Runge--Kutta (RK) methods. In particular, we construct a new class of conservative finite difference methods by applying weighted essentially nonoscillatory (WENO) reconstructions to the so-called time-averaged fluxes. Our schemes are automatically conservative under any modification of the fluxes, which is attributed to the fact that classical WENO reconstructions conserve mass when coupled with forward Eu...

Field-aligned semi-Lagrangian methods for turbulence simulations of strongly magnetized plasmas

Summary form only given. Plasma turbulence in strongly magnetized plasmas is characterized by long wavelengths along the magnetic field, of the order of the system size, and by short perpendicular wavelengths, of the order of the ion gyro-radius. This strong anisotropy suggests the use of field-aligned coordinates in order to drastically reduce the number of grid points along a certain direction (usually parallel to the field).Traditionally, field-aligned discretizations are derived from predefined flux coordinates; in practice, this limits their use to simple geometries and magnetic field configurations. Recently [1] the use of more general field-aligned coordinates was proposed, which do not depend on flux surface variables. This approach allows for increased flexibility, and it is suitable to modeling X-point magnetic field configurations. Therefore, it is a perfect candidate for the global simulation of modern magnetic fusion reactors like ITER. Here we describe the implementation of this new field-aligned strategy within the semi-Lagrangian library SeLaLib [2], for solution of the gyrokinetic equations in 5D phase-space [3]. Our aim is to simulate turbulence in Tokamak and Stellarator reactors. We present verification of our code against analytical solutions for a periodic cylinder, as well as preliminary results in axially-symmetric toroidal geometry (Tokamak). Finally, we discuss the future extension of the code to more general toroidal geometries (Stellarator).

Global model capability study of EEDF modification of rare gas metastable laser reaction kinetics

Extending from revived interest in the study of diode-pumped alkali vapor lasers (DPAL), it was shown that optically pumping a rare gas metastable state can result in a population inversion with similar spectral characteristics to those making DPAL attractive1. Both systems can be pumped incoherently resulting in a temporally coherent output while a rare gas laser (RGL) does not suffer the extremely reactive behavior of alkali metals. Metastable species are produced under electric discharge and are relatively inert with respect to buffer gases and system construction. We propose using controlled electron energy distributions (EEDF) to modify RGL efficiency and to potentially drive the gain mechanism without the need for intense optical pumping. Formation of the EEDF is dependent on electric discharge conditions and introduction of electron sources.

A higher order a-stable Maxwell solver using successive convolution

Summary form only given. We develop a numerical scheme for solving Maxwells equations to high accuracy in both space and time, without incurring the traditional Courant-Friedrichs-Lewy (CFL) stability limit on the time step. Our scheme utilizes a novel technique, successive convolution, to achieve order 2P in time at N spatial points, with a cost of O(PdN) in d spatial dimensions. The speed and efficiency of the solver is due to the dimensional splitting we employ, which means that spatial convolution is performed line-by-line, making it logically Cartesian. However, the spatial points can be adjusted without affecting the stability, and so we can freely embed smooth non-rectangular domains into the grid, thereby preserving accuracy.

General-purpose kinetic global modeling framework for multi-phase chemistry

Summary form only given. Spatially averaged (global) models are ubiquitous in plasma science, and the required data and equations are conceptually very similar for most applications. Unfortunately, it is common practice to implement a custom-developed software for each new global model; this unnecessary duplication of efforts negatively affects quality control and code maintenance. We present a general purpose kinetic global modeling framework (KGMf) designed to support plasma scientists in all modeling phases: collection and analysis of the reaction data, automatic construction of a system of ordinary differential equations (ODEs), time evolution of the system, and dynamical optimization of some target function. Alongside the description of the software, we present a few example tutorials.

Conservative and positivity-preserving semi-Lagrangian kinetic schemes with spectrally accurate phase-space resolution

Summary form only given. The Convected Scheme (CS) is a family of semi-Lagrangian algorithms, most usually applied to the solution of Boltzmanns equation, which uses a method of characteristics in an integral form to project a moving cell (MC) forward to a group of mesh cells. In earlier work [1], a 4th-order version of the cell-centered CS was presented, which was based on applying an a-priori correction to the position of the MC after the ballistic move and prior to remapping to the mesh. Such corrections were calculated by means of a modified equation analysis applied to the continuity equation with a prescribed flow field. The resulting 4th-order CS showed a drastically reduced numerical diffusion, while it retained the desirable properties of the original scheme (i.e. mass conservation, positivity preservation, and simplicity). In this contribution we describe higher order versions of the CS, suited to the accurate solution of the Vlasov equation with minimum computational resources. By applying an appropriate operator splitting procedure, the solution to the Vlasov-Poisson (or Vlasov-Maxwell) system can be reduced to a succession of constant advection steps, either in configuration or in velocity space, interleaved with appropriate field updates. With this setting in mind, we specialize our analysis to the constant advection equation, and we illustrate a new procedure that extends the CS to arbitrarily high order of accuracy. We describe a nominally 22nd-order CS, in which we compute the required 20 spatial derivatives of the solution using a fast Fourier transform. For smooth profiles, this scheme shows spectral convergence to the exact solution, and hence we refer to it as “Spectral CS”. Further, adaptive filtering in Fourier space permits us to resolve non smooth profiles without introducing spurious oscillations. We show the schemes behavior in typical 1D-1V test cases for the Vlasov-Poisson system, both in periodic and bounded domains, with one or more species. We then discuss higher dimensional problems, where the computational savings inherent to the Spectral CS would enable unprecedented phase-space resolution. Finally we consider the solution of the Vlasov-Maxwell system, as well as the inclusion of collisional processes.

Non-equilibrium kinetics of a microwave-assisted jet flame: Global model and comparison with experiment

Developing a systematic understanding of plasma driven chemical reaction pathways is difficult due to the stiffness and non-linear processes inherent with the involved physics. In the context of microwave-coupled plasmas within atmospheric pressure nozzle geometries, we have developed a kinetic global model (KGM) framework designed for quick exploration of parameter space. Our final goal is understanding key reaction pathways within non-equilibrium plasma assisted combustion (PAC), and their roles in the combustion process; of primary importance is the ability to determine possible system dependent reaction mechanism augmentation and specific reaction selectivity. In combination with a Boltzmann equation solver, kinetic plasma and gas-phase chemistry are coupled with a compressible gas flow model and solved with iterative feedback to match observed bulk conditions from experiments. We use a non-equilibrium electron energy distribution function (EEDF) to define electron-impact processes, allowing for demonstration of variation in reaction pathways due to changes in the EEDF shape. An Eulerian approach was developed as a purely steady-state flow model, followed by a Lagrangian time-dependent approach requiring knowledge of spatial electromagnetic field and flow profiles. Spatial profiles are converted into time-dependent envelopes affecting system parameters. The KGM is first applied to argon and air (N2-O2, N2-O2-Ar) systems as a means of assessing the soundness of the assumptions inherent in any global model. The simplified nature of the gas-phase chemistry and the availability of cross-sectional data reduce the sources of uncertainty in the model, which can be validated against the experimental measurements of electron density, emission spectrum and gas temperature. The test with air greatly increases the complexity by incorporating a plethora of excited states, providing new energy sink mechanisms (e.g. translational and vibrational excitation) and reaction pathways. The KGM is then applied to plasma driven combustion mechanisms (e.g. H2 or CH4 with an oxidizer source), which increases the importance of flow treatment and the range of gas-phase chemistry time-scales. As the reaction mechanisms become more complex, the limits of available data will begin to hinder model physicality, requiring analytical and/or empirical treatment of gaps in data to maintain completeness of the reaction mechanisms. Due to the relative simplicity of simulations with global models, the KGM can also be used to provide a sensitivity analysis to errors and variations within available data.