Abstract
A useful approach for long range computation of the Helmholtz equation in a waveguide is to re-formulate it as the operator differential Riccati equation for the Dirichlet-to-Neumann map. For waveguides with slow range dependence, the piecewise range-independent approximation is used to derive a second-order range stepping method for this one-way re-formulation. The range marching formulas are exact for each range-independent piece and a large range step is possible if the range dependence is gradual. Based on a fourth-order conservative exponential method for linear evolution equations, a fourth-order method that admits even larger range steps is developed for the one-way re-formulation. Numerical examples are used to demonstrate the improved accuracy of the fourth-order method.
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