Adaptive-grid methods with asymmetric time-filtering

Abstract

We identify the three essential parts for a successful and robust adaptive grid distribution equation for systems of nonlinear partial differential equations: (1) the spatial function, which is composed of operators that control the shape and size of zones, and guarantees the stability and positivity of the corresponding metric, (2) the structure function, which provides an objective measure of the structure of the solution, and generates source terms that drive the grid distribution into any desired configuration, and (3) an asymmetric time filter, which ensures a smooth space-time evolution of the mesh globally, even when there are abrupt changes in the structure of the solution locally. The last part ensures that any catastrophic effects of grid redistribution due to the sudden appearance or disappearance of discontinuities in the solution caused by nonlinear interactions are avoided. Specifically, we describe and analyze an implicit, finite-difference, adaptive-grid technique for time dependent (as well as time independent) problems in one spatial dimension, which automatically detects, resolves and tracks all essential features in the solution. Our method is demonstrated on two examples in fluid dynamics: (1) the generation and propagation of a front in two-phase immiscible flow described by the Buckley-Leverett equation, and (2) the classical Riemann problem of ideal gas dynamics.