Conservative front tracking and level set algorithms

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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.

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