The exponential method of angled derivatives

A hybrid numerical method is presented for a linear, first order, hyperbolic partial differential equation (PDE): the nonhomogeneous one-way advection-reaction equation. This PDE is first reduced to an ordinary differential equation (ODE) along a characteristic emanating backward in time from each mesh point. These ODEs are then integrated numerically using exponential fitting to advance in time thus permitting wide disparities in size among the coefficients. Such temporal integration requires a spatial interpolation procedure dependent upon the slope of the local characteristic. If the local Courant number is less than one then the interpolation is afforded by the exponential angled derivative method while if greater than one then the exponential reflected angled derivative approximation is employed. Otherwise, if the Courant number is transitioning through one then the exponential box scheme is employed. At the boundary, parabolic tracing of the characteristics provides a comparable level of accuracy. The net result is a three-level explicit method which is second-order accurate and unconditionally stable. The efficacy of this exponential method of angled derivatives (EMAD) scheme for singular perturbation problems is demonstrated numerically.