ACM Transactions on Mathematical Software (TOMS)
Accurate finite difference methods for time-harmonic wave propagation
Journal of Computational Physics
ACM Transactions on Mathematical Software (TOMS)
Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Numerical solution of problems on unbounded domains. a review
Applied Numerical Mathematics - Special issue on absorbing boundary conditions
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions
Journal of Computational Physics
A class of difference schemes with flexible local approximation
Journal of Computational Physics
Compact finite difference schemes of sixth order for the Helmholtz equation
Journal of Computational and Applied Mathematics
A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media
Journal of Computational Physics
The weak element method applied to Helmholtz type equations
Applied Numerical Mathematics
Compact optimal quadratic spline collocation methods for the Helmholtz equation
Journal of Computational Physics
Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number
Journal of Computational Physics
Journal of Computational Physics
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In many problems, one wishes to solve the Helmholtz equation in cylindrical or spherical coordinates which introduces variable coefficients within the differentiated terms. Fourth order accurate methods are desirable to reduce pollution and dispersion errors and so alleviate the points-per-wavelength constraint. However, the variable coefficients renders existing fourth order finite difference methods inapplicable. We develop a new compact scheme that is provably fourth order accurate even for these problems. The resulting system of finite difference equations is solved by a separation of variables technique based on the FFT. Moreover, in the r direction the unbounded domain is replaced by a finite domain, and an exact artificial boundary condition is specified as a closure. This global boundary condition fits naturally into the inversion of the linear system. We present numerical results that corroborate the fourth order convergence rate for several scattering problems.