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We present a numerical method for computing the solution of a partialdifferential equation (PDE) for modeling acoustic pressure, known as anextra-wide angle parabolic equation, that features the square root of adifferential operator. The differential operator is the negative of anindefinite Schr\"{o}dinger operator with a piecewise constant potential. Thiswork primarily deals with the 3-piece case; however, a generalization is madethe case of an arbitrary number of pieces. Through restriction to a judiciouslychosen lower-dimensional subspace, approximate eigenfunctions are used toobtain estimates for the eigenvalues of the operator. Then, the estimatedeigenvalues are used as initial guesses for the Secant Method to find the exacteigenvalues, up to roundoff error. An eigenfunction expansion of the solutionis then constructed. The computational expense of obtaining each eigenpair isindependent of the grid size. The accuracy, efficiency, and scalability of thismethod is shown through numerical experiments and comparisons with othermethods.

Numerical Solution of an Extra-wide Angle Parabolic Equation through  Diagonalization of a 1-D Indefinite Schr\"{o}dinger Operator with a Piecewise  Constant Potential