1,486 research outputs found
CASTRO: A New Compressible Astrophysical Solver. II. Gray Radiation Hydrodynamics
We describe the development of a flux-limited gray radiation solver for the
compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with
block-structured adaptive mesh refinement based on a nested hierarchy of
logically-rectangular variable-sized grids with simultaneous refinement in both
space and time. The gray radiation solver is based on a mixed-frame formulation
of radiation hydrodynamics. In our approach, the system is split into two
parts, one part that couples the radiation and fluid in a hyperbolic subsystem,
and another parabolic part that evolves radiation diffusion and source-sink
terms. The hyperbolic subsystem is solved explicitly with a high-order Godunov
scheme, whereas the parabolic part is solved implicitly with a first-order
backward Euler method.Comment: accepted for publication in ApJS, high-resolution version available
at https://ccse.lbl.gov/Publications/wqzhang/castro2.pd
Blowup in diffusion equations: A survey
AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems
On the Computation of Blow-up Solutions for Semilinear ODEs and Parabolic PDEs
We introduce an adaptive numerical method for computing blow-up solutions for
ODEs and well-known reaction-diffusion equations. The method is based on the
implicit midpoint method and the implicit Euler method. We demonstrate that
the method produces superior results to the adaptive PECE-implicit method
and the MATLAB solver of comparable order
A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System
We introduce an efficient stochastic interacting particle-field (SIPF)
algorithm with no history dependence for computing aggregation patterns and
near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis
system in three space dimensions (3D). The KS solutions are approximated as
empirical measures of particles coupled with a smoother field (concentration of
chemo-attractant) variable computed by the spectral method. Instead of using
heat kernels causing history dependence and high memory cost, we leverage the
implicit Euler discretization to derive a one-step recursion in time for
stochastic particle positions and the field variable based on the explicit
Green's function of an elliptic operator of the form Laplacian minus a positive
constant. In numerical experiments, we observe that the resulting SIPF
algorithm is convergent and self-adaptive to the high gradient part of
solutions. Despite the lack of analytical knowledge (e.g. a self-similar
ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study
the emergence of finite time blowup in 3D by only dozens of Fourier modes and
through varying the amount of initial mass and tracking the evolution of the
field variable. Notably, the algorithm can handle at ease multi-modal initial
data and the subsequent complex evolution involving the merging of particle
clusters and formation of a finite time singularity
A moving mesh method for one-dimensional hyperbolic conservation laws
We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work
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