7,155 research outputs found
A Hybrid Godunov Method for Radiation Hydrodynamics
From a mathematical perspective, radiation hydrodynamics can be thought of as
a system of hyperbolic balance laws with dual multiscale behavior (multiscale
behavior associated with the hyperbolic wave speeds as well as multiscale
behavior associated with source term relaxation). With this outlook in mind,
this paper presents a hybrid Godunov method for one-dimensional radiation
hydrodynamics that is uniformly well behaved from the photon free streaming
(hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and
to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds
that the technique preserves certain asymptotic limits. The method incorporates
a backward Euler upwinding scheme for the radiation energy density and flux as
well as a modified Godunov scheme for the material density, momentum density,
and energy density. The backward Euler upwinding scheme is first-order accurate
and uses an implicit HLLE flux function to temporally advance the radiation
components according to the material flow scale. The modified Godunov scheme is
second-order accurate and directly couples stiff source term effects to the
hyperbolic structure of the system of balance laws. This Godunov technique is
composed of a predictor step that is based on Duhamel's principle and a
corrector step that is based on Picard iteration. The Godunov scheme is
explicit on the material flow scale but is unsplit and fully couples matter and
radiation without invoking a diffusion-type approximation for radiation
hydrodynamics. This technique derives from earlier work by Miniati & Colella
2007. Numerical tests demonstrate that the method is stable, robust, and
accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61
pages, 15 figures, 11 table
Modelling binary alloy solidification with adaptive mesh refinement
The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
An exact particle method for scalar conservation laws and its application to stiff reaction kinetics
An "exact" method for scalar one-dimensional hyperbolic conservation laws is
presented. The approach is based on the evolution of shock particles, separated
by local similarity solutions. The numerical solution is defined everywhere,
and is as accurate as the applied ODE solver. Furthermore, the method is
extended to stiff balance laws. A special correction approach yields a method
that evolves detonation waves at correct velocities, without resolving their
internal dynamics. The particle approach is compared to a classical finite
volume method in terms of numerical accuracy, both for conservation laws and
for an application in reaction kinetics.Comment: 14 page, 7 figures, presented in the Fifth International Workshop on
Meshfree Methods for Partial Differential Equation
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