Conservative front tracking and level set algorithms

Abstract

Hyperbolic conservation laws are foundational for many branches of continuum physics. Discontinuities in the solutions of these partial differential equations are widely recognized as a primary difficulty for numerical simulation, especially for thermal and shear discontinuities and fluid-fluid internal boundaries. We propose numerical algorithms that will (i) track these discontinuities as sharp internal boundaries, (ii) fully conserve the conserved quantities at a discrete level, even at the discontinuities, and (iii) display one order of numerical accuracy higher globally (at the discontinuity) than algorithms in common use. A significant improvement in simulation capabilities is anticipated through use of the proposed algorithms.H yperbolic conservation laws are foundational for many branches of continuum physics. Discontinuities in the solutions of these partial differential equations are widely recognized as a primary difficulty for numerical simulation. Commonly used numerical algorithms are convergent at some power of the mesh spacing ⌬x, but only for smooth solutions. At solution discontinuities, the local truncation errors are typically 0 (1), e.g., not convergent. For nonlinear shock waves, this situation is mitigated by the fact that errors flow into the discontinuity (which functions as a ''black hole'' in this regard), and the size of the error region does not grow. For linear-type discontinuities, such as thermal or fluid material boundaries, the discontinuities spread and often occupy 5-10 mesh cells as the simulation evolves into late time. The ability to solve accurately many practical problems is hampered by these facts.Front tracking was proposed as a (partial) cure for these problems (1-3). The method has recently been extended to three dimensions and given a robust and simple interface description (4-6). In this method, a sharp numerical boundary is maintained within the computation to prevent the artificial mixing of fluids across a fluid interface. The communication of information across the interface is accomplished by use of analytic solutions of idealized jump discontinuities (Riemann problems) and by ghost cells, to maintain data extrapolated across the interface. The ghost cells are needed by the finite difference operators approximating the differential equation. The ghost-cell extrapolation method was introduced in 1981 by Glimm, Marchesin, and McBryan (3) for front tracking. This method, and a closely related ghost-cell level set method (7) proposed in 1999, are only partial solutions to the problem of simulation of fluid interfaces. Both lack conservation in the cells cut by the interface and both have only conventional accuracy with 0 (1) local truncation errors at the discontinuities. Thus, neither is a correct solution of the interface problem.The purpose of this paper is to propose a tracking͞level set algorithm that is conservative even at discontinuities and that improves by one order of accuracy over conventional algorithms. so called because ͐ udx changes in time only bec...

Interface tracking method for compressible multifluids

“…They are easy to implement in the one-dimensional (1-D) case, but far less trivial to implement in several space dimensions [7,[12][13][14]44]. Moving mesh techniques have been proposed as a way of keeping solution discontinuities sharp [20].…”

Terascale Simulation Tools and Technologies

“…Then we calculate the fluxes defined on the surfaces of V i , so that we can apply (4). Works on this important improvement of the front tracking method have been published in [25,32,33].…”

A Second-Order Accurate Conservative Front-Tracking Method in One Dimension

“…Another approach is cell-merging. It was used by Glimm, Li, and Zhao [10], where a second-order accurate algorithm on the one-dimensional Burgers's equation is presented, and in Glimm et al [9], where the method is extended to multiple dimensions. In cell-merging, the small control volumes are merged with adjacent cells to avoid singularities, but, while finding a full cell for each small cell is a well-behaved problem in one dimension, it is not always straightforward to do in multiple dimensions.…”

“…The details of the redistribution algorithm are given in section 2.4.3. In one dimension, redistribution could have easily been avoided by using cell-merging (see, e.g., [10]). However, cell-merging in three dimensions leads to difficult geometric constructions that have not been fully worked out, and it is not clear how to combine them with discretization methods that maintain O(h) truncation error near the embedded boundary.…”