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As a follow up to Causley2013, we provide a detailed description of the numerical implementation of an O(N), A-stable, second order accurate solution of the wave equation, constructed from semi-discrete boundary value problems. We improve on the previous algorithm by replacing the Lax-type correction used in Causley2013, which was necessary for convergence when Δ t < Δ x/c, with a more accurate spatial quadrature, which we prove is convergent. We also demonstrate that the resulting solverarXiv:1306.6902v1 fatcat:tbtkbwlwtbfoncmoa3x4bmrvsm