Georgia Institute of Technology College of Sciences

In this dissertation, we studied the Back and Forth Error Compensation and Correction (BFECC) method for linear hyperbolic PDE systems and nonlinear scalar conservation laws. We extend the BFECC method from scalar hyperbolic PDEs to linear hyperbolic PDE systems, and showed similar stability and accuracy improvement are still valid under modest assumptions on the systems. Motivated by this theoretical result, we propose BFECC schemes for the Maxwell's equations. On uniform orthogonal grids, the BFECC schemes are guaranteed to be second order accurate and have larger CFL numbers than that of the classical Yee scheme. On non-orthogonal and unstructured grids, we propose to use a simple least square local linear approximation scheme as the underlying scheme for the BFECC method. Numerical results showed the proposed schemes are stable and are second order accurate on non-orthogonal grids and for systems with variable coefficients. We also studied a conservative BFECC limiter that reduces spurious oscillations for numerical solutions of nonlinear scalar conservation laws. Numerical examples with the Burgers' equation and KdV equations are studied to demonstrate effectiveness of this limiter.