A Cartesian grid embedded boundary method for hyperbolic conservation laws

doi:10.1016/j.jcp.2005.05.026

Journal of Computational Physics

Volume 211, Issue 1, 1 January 2006, Pages 347-366

https://doi.org/10.1016/j.jcp.2005.05.026 Get rights and content

Abstract

We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L¹ for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.

Introduction

In this paper, we present an extension of the unsplit method for time-dependent hyperbolic conservation laws in [8], [17] to the case of an embedded boundary representation of irregular geometries. Our approach is a generalization of the conservative method in [14], following the ideas in [2], [6]. The present method uses the approach in [9] to construct a method that is formally consistent. Near the embedded boundary, the truncation error is O(h), while in the interior, the truncation error is O(h²), where h is the mesh spacing. This leads to a method for which the solution error is O(h²) in L¹, and O(h) in L^∞. This differs from the methods cited above, which are formally inconsistent, i.e., have O(1) truncation errors near the embedded boundary. Our method uses a linear hybridization of explicit conservative and nonconservative updates combined with the use of the flux redistribution ideas in [2], [6], [14] to maintain local conservation. We present results for smooth solutions in two and three space dimensions that verify that the accuracy of the method is consistent with the truncation error and modified equation analysis. We also present results for strong shock problems that demonstrate the robustness and accuracy of the method. A preliminary version of this work appears in [13].

Section snippets

Notation

Cartesian grids with embedded boundaries are useful to describe finite-volume representations of solutions to PDE in the presence of irregular boundaries. In Fig. 1, the gray area represents the region excluded from the solution domain. The underlying description of space is given by rectangular control volumes on a Cartesian grid $ϒ_{i} = [(i - \frac{1}{2} v) h, (i + \frac{1}{2} v) h], i \in Z^{D}$ , where D is the dimensionality of the problem, h is the mesh spacing, and $v$ is the vector whose entries are all one. Given an irregular

Stable evolution of hyperbolic conservation laws

We want to solve a hyperbolic system of conservation laws $\frac{\partial U}{\partial t} + \nabla \cdot \vec{F} = 0,$ $U = U (x, t), x \in Ω \subset R^{D},$ $(F^{1} \dots F^{D}) = \vec{F} = \vec{F} (U),$ $U, F^{d} \in R^{m} .$ While the algorithm we describe here applies to general systems of hyperbolic conservation laws, we will show results for the case of polytropic gas dynamics in two and three dimensions $U = (ρ, ρ u, ρ E)^{T},$ $F^{d} (U) = (ρ u^{d}, ρ u^{d} u + p e^{d}, ρ u^{d} E + u^{d} p) .$ Here $ρ$ is the fluid density, $(u^{1} \dots u^{D}) = u \in R^{D}$ is the velocity, $E \equiv \frac{p}{(γ - 1) ρ} + \frac{1}{2} | u |^{2}$ the total energy per unit mass, and p the pressure and $γ > 1$ the ratio of specific

Flux calculation

Given $U_{i}^{n}$ , we need to compute an O(h²) estimate of the fluxes $F_{i + \frac{1}{2} e^{d}}^{n + \frac{1}{2}} \approx F^{d} ((i + \frac{1}{2} e^{d}) h, t^{n} + \frac{1}{2} Δ t) .$ Specifically, we want to compute the fluxes at the center of the Cartesian grid faces corresponding to the faces of the embedded boundary geometry.

For many applications it is useful to perform the flux calculation using nonconservative variables W = W(U). For the case of polytropic gas dynamics, the primitive variables are W = (ρ, u, p)^T. The quasilinear equations for these variables are given as follows: $\frac{\partial W}{\partial t} + \sum d$

Slope calculation

The notation $CC = A | B | C$ means that the 3-point formula A is used for CC if all cell-centered values it uses are available, the 2-point formula B is used if the cell to the right (i.e., the high side) of the current cell is covered, and the 2-point formula C is used if the cell to the left (i.e., the low side) current cell is covered.

To compute the limited differences in the first step on the algorithm, we use the fourth-order slope calculation [8] combined with characteristic limiting $\begin{matrix} Δ_{4}^{d} W_{i} = ζ_{i} {\tilde{Δ}}_{4}^{d} W_{i}, \end{matrix}$

Convergence tests

Our test problem is a simple wave propagating in a straight circular channel (a straight-walled channel in two dimensions). The flow field is a stagnant fluid with a small perturbation in a single Riemann invariant. We specify an initial profile for density at time $t = 0$ , $ρ_{0} (x) = ρ_{ref} (1 + α f (\bar{x})),$ where $f (\bar{x}) = \{\begin{matrix} ({\bar{x}}^{2} - 1)^{4} & if 0 ⩽ \bar{x} ⩽ 1, \\ 0 & otherwise \end{matrix}$ with the dimensionless coordinate $\bar{x} = \vec{x} \cdot \hat{n} / w .$ The parameters are: $α$ , the amplitude of the wave; w, the width of the wave; and $\hat{n}$ , the direction of propagation of the wave.

Conclusions

We presented here a new Cartesian grid embedded boundary algorithm for systems of conservation laws, generalizing the unsplit second-order Godunov method described in [8], [17]. It is formally consistent, with a truncation error that vanishes as the mesh spacing goes to zero, leading to a method that is at least second-order accurate in L¹, and first-order accurate in L^∞. On standard strong-shock test problems, it is robust, leading to results that are nearly indistinguishable from

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¹: Research supported at the Lawrence Berkeley National Laboratory by the US Department of Energy: Director, Office of Science, Office of Advanced Scientific Computing, Mathematical, Information, and Computing Sciences Division under Contract DE-AC03-76SF00098.

²: Research supported by the Computational Science Graduate Fellowship program of the Department of Energy, under Grant No. DE-FG02-97ER25308.

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A Cartesian grid embedded boundary method for hyperbolic conservation laws

Abstract

Introduction

Section snippets

Notation

Stable evolution of hyperbolic conservation laws

Flux calculation

Slope calculation

Convergence tests

Conclusions

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.