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Stable high-order finite-difference interface schemes with application to the Richtmyer-Meshkov instability

Citation

Kramer, Richard (2009) Stable high-order finite-difference interface schemes with application to the Richtmyer-Meshkov instability. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-03132009-095507

Abstract

High-order adaptive mesh refinement offers the potential for accurate and efficient resolution of problems in fluid dynamics and other fields where a wide range of length scales is present. A critical requirement for the interface closures used with these methods is stability in the context of hyperbolic systems of partial differential equations.

In this study, a class of energy-stable high-order finite-difference interface closures is presented for grids with step resolution changes in one dimension. Asymptotic stability in time for these schemes is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface closures compatible with interior fourth- and sixth-order explicit, and fourth-order implicit centered schemes are presented. Validation tests include linear and nonlinear problems in one and in two dimensions with tensor-product grid refinement.

A second class of stable high-order interface closures is presented for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Stability is achieved by again imposing a summation-by-parts condition, resulting in nondissipative closures, at the cost of accuracy at corner interfaces. Interface stencils for an explicit fourth-order finite-difference scheme are presented for each geometry. Validation tests confirm the stability and accuracy of these closures for linear and nonlinear problems.

The Richtmyer-Meshkov instability is investigated using a novel first-order perturbation of the two-dimensional Navier-Stokes equations about a shock-resolved base flow. The computational domain is efficiently resolved using the one-dimensional fourth-order interface scheme. Results are compared to analytic models of the instability, showing agreement with predicted asymptotic growth rates in the inviscid range, while significant discrepancies are noted in the transient growth phase. Viscous effects are found to be poorly predicted by existing models.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:interface schemes; mesh refinement; shock resolution
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Aeronautics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Pullin, Dale Ian
Thesis Committee:
  • Pullin, Dale Ian (chair)
  • Leonard, Anthony
  • Meiron, Daniel I.
  • Shepherd, Joseph E.
Defense Date:11 March 2009
Record Number:CaltechETD:etd-03132009-095507
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-03132009-095507
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:947
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:13 May 2009
Last Modified:26 Dec 2012 02:33

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