Dynamically Bi-orthogonal Field Equations for Solution of Two-Dimensional Hyperbolic Partial Differential Equations

摘要

This paper explores solution of two-dimensional hyperbolic partial differential equations using dynamically bi-orthogonal field equations (DBFE). Application of stochastic spectral methods for solution of hyperbolic partial differential equations result in evolution of Gibbs phenomenon, which is characterized by the oscillations near discontinuities. The DBFE method uses spectral expansion of spatial dimension in eigenfunction basis, while the stochastic dimension is decomposed in generalized polynomial chaos basis. Post-processing is used on the resultant solution to mitigate the effect of the Gibbs phenomenon. This paper investigates the DBFE method for solution of a two-dimensional Burgers equation. Existence of the Gibbs phenomenon in the DBFE solution and ability of the post-processing approach to mitigate the effect in two-dimensions is demonstrated.