A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the $C$-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution $C(x,t)$ is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the $C$-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our $C$-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure

A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines approach combining discontinuous Galerkin spatial discretization with single- or multi-step methods in time. The construction of the estimators requires a reconstruction in time for which we present a very general framework first for odes and then apply the approach to conservation laws. The reconstruction does not depend on the actual method used for evolving the solution in time. Most importantly it covers in addition to implicit methods also the wide range of explicit methods typically used to solve conservation laws. For the spatial discretization, we allow for standard choices of numerical fluxes. We use reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. We study under which conditions on the numerical flux the estimate is of optimal order pre-shock. While the estimator we derive is computable and valid post-shock for fixed meshsize, it will blow up as the meshsize tends to zero. This is due to a breakdown of the relative entropy framework when discontinuities develop. We conclude with some numerical benchmarking to test the robustness of the derived estimator

Moment-Based Accelerators for Kinetic Problems with Application to Inertial Confinement Fusion

In inertial confinement fusion (ICF), the kinetic ion and charge separation field effects may play a significant role in the difference between the measured neutron yield in experiments and the predicted yield from fluid codes. Two distinct of approaches exists in modeling plasma physics phenomena: fluid and kinetic approaches. While the fluid approach is computationally less expensive, robust closures are difficult to obtain for a wide separation in temperature and density. While the kinetic approach is a closed system, it resolves the full 6D phase space and classic explicit numerical schemes restrict both the spatial and time-step size to a point where the method becomes intractable. Classic implicit system require the storage and inversion of a very large linear system which also becomes intractable. This dissertation will develop a new implicit method based on an emerging moment-based accelerator which allows one to step over stiff kinetic time-scales. The new method converges the solution per time-step stably and efficiently compared to a standard Picard iteration. This new algorithm will be used to investigate mixing in Omega ICF fuel-pusher interface at early time of the implosion process, fully kinetically

Numerical modelling of entropy production in mixed convection heat transfer

Energy losses in fluids engineering systems occur due to thermal and viscous irreversibilities. These irreversibilities can be tracked to identify regions of design modification for efficiency improvement in thermofluid systems. The rate of entropy production in numerical heat transfer is an important parameter that characterizes the degree of these irreversibilities. This can lead to improved designs with higher system efficiency levels for energy savings in various engineering applications. Previous conventional techniques have generally detected energy losses on a global scale or end-to-end basis. This thesis focuses on two-dimensional numerical modeling of entropy generation and the Second Law of Thermodynamics in mixed convection heat transfer. A Control-Volume Based Finite Element Method (CVFEM) is used to discretize and solve the governing conservation equations. An entropy-based algorithm is developed by post-processing of the velocity and temperature fields to obtain numerical predictions of the rate of entropy production. The new model is used to analyze heat transfer and entropy production for both natural and mixed convection in enclosures filled with different fluids, including nanofluids. The optimal conditions for which viscous and thermal irreversibilities are minimized is analyzed. The results from Computational Fluid Dynamics (CFD) are validated using available benchmark data. A new approach for minimizing the rate of entropy production in different flow configurations with nanofluids is also obtained. In addition, the local entropy production rates are obtained from two forms of the discretized Second Law – namely, transport and positive-definite forms of the entropy transport equation. The computed local entropy generation rates from both methods are compared and related to expected numerical errors from available benchmark solutions. An entropy-based error indicator is determined to assess the solution accuracy of fluid flow simulations with heat transfer using the Second Law of Thermodynamics. The formulation presents a new approach for the characterization of numerical error using a parameter called the “apparent entropy production difference.” Furthermore, a corrective mechanism on the numerical algorithm is developed. The transport entropy is used to calculate an artificial viscosity (named as an entropy-based artificial viscosity) to reduce the numerical error and ensure closer compliance with the Second Law

Extension of the Entropy Viscosity Method to the Multi-D Euler Equations and the Seven-Equation Two-Phase Model

The work presented in this dissertation focuses on the application of the entropy viscosity method to low-Mach single- and two-phase flow equations discretized using a continuous Galerkin finite element method with implicit time integration. The technique has been implemented and tested using the multiphysics simulation environment MOOSE (D Gaston, C Newsman, G Hansen and D Lebrun-Grandie. A parallel computational framework for coupled systems of nonlinear equations. Journal of Nucl. Eng. Design, 239, 1768-1778, 2009). First, the entropy viscosity method, developed by Guermond et al. (J-L Guermond, R Pasquetti and B Popov. Entropy viscosity method for nonlinear conservation laws. Journal of Comput. Phys., 230, 4248-4267, 2011), is extended to the multi-dimensional Euler equations for both subsonic (very low Mach numbers) and supersonic flows. We show that the current definition of the viscosity coefficients is not adapted to low-Mach flows and we provide a robust alternate definition valid for any Mach number value. The new definitions are derived from a low-Mach asymptotic study, is valid for a wide range of Mach numbers and no longer requires an analytical expression of the entropy function. In addition, the entropy minimum principle is used to derive the viscous regularization terms for Euler equations with variable area for nozzle flow problems and was proved valid for any equation of state with a concave entropy. The new definition of the entropy viscosity method is tested on various 1-D and 2-D numerical benchmarks employing the ideal and the stiffened gas equation of states: flow in a converging-diverging nozzle, Leblanc shock tube, slow moving shock, strong shock for liquid phase, subsonic flows around a 2-D cylinder and over a circular hump, and supersonic flow in a compression corner. Convergence studies are performed using analytical solutions in 1-D and proved the entropy viscosity method to be second-order accurate for smooth solutions. In a second part, the entropy viscosity method is applied to the seven-equation two-phase flow model. After deriving the dissipative terms using the same procedure as for the multi-D Euler equations, a low-Mach asymptotic study is performed in order to obtain a definition for the viscosity coefficients. Because the seven-equation model is derived by assuming that each phase obeys the Euler equations, the dissipative terms and the definition of the viscosity coefficients are analogous to the ones obtained for the single-phase system of equations. Then, 1-D numerical tests were performed to demonstrate that the entropy viscosity method properly stabilizes the flow simulations based on the seven-equation model. Another focus of this work was to investigate the impact of source terms (gravity, friction, etc) onto the entropy viscosity method. The theoretical approach adopted here consists of deriving the entropy residual when accounting for the source terms and investigate the sign of the new terms in order to adapt the definition of the viscosity coefficients. Numerical 1-D tests are performed to validate this approach for both single- and two-phase flow models. In the last part of this dissertation, the entropy viscosity method is applied to the 1-D grey radiation-hydrodynamic equations where the 1-D Euler equations are coupled to a radiation diffusion equation through relaxation terms. The method of manufactured solutions was used to prove second-order accuracy of the numerical stabilization method and also show that the entropy viscosity method yields the correct asymptotic diffusion limit. 1-D tests for inlet Mach number ranging from 1.2 to 50 are presented and show good agreement with semi-analytical solutions