An Adaptive Finite Difference Method for Hyperbolic Systems on One Space Dimension, Revision,
CALIFORNIA UNIV BERKELEY LAWRENCE BERKELEY LAB
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In this paper we develop and partially analyze an adaptive finite difference mesh refinement algorithm for the initial boundary value problem for hyperbolic systems in one space dimension. The method uses clusters uniform grids which can move along with pulses or steep gradients appearing in the calculation, and which are superimposed over a uniform coarse grid. Such refinements are created, destroyed, merged, separated, recursively nested or moved based on estimates of the local truncation error. We use a four-way linked tree and sequentially allocated deques double-ended queues to perform these operations efficiently. The local truncation error is estimated using a three-step Richardson extrapolation procedure in the interior of the region, and differences at the boundaries. Our algorithm was implemented using a portable, extensible Fortran preprocessor, to which we added records and pointers. The method is applied to two model problems the second order wave equation with counterstreaming Gaussian pulses, and the Riemann shock-tube problem. For both problems our algorithm is shown to be three to five times more efficient in computing time than the use of a uniform coarse mesh, for the same accuracy.
- Theoretical Mathematics