Error analysis for the operator marching method applied to range dependent waveguides

The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize "approximation level" to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis.