I. Singer et E. Turkel, HIGH-ORDER FINITE-DIFFERENCE METHODS FOR THE HELMHOLTZ-EQUATION, Computer methods in applied mechanics and engineering, 163(1-4), 1998, pp. 343-358
High-order finite difference methods for solving the Helmholtz equatio
n are developed and analyzed, in one and two dimensions on uniform gri
ds. The standard pointwise representation has a second-order accurate
local truncation error. We also study two schemes which have a fourth-
order accurate local truncation error. One of the high-order schemes i
s based on generalizations of the Pade approximation. The second schem
e is based on high-order approximation to the derivative calculated fr
om the Helmholtz equation itself. A symmetric high-order representatio
n is developed for a Neumann boundary condition. Numerical results are
presented on model problems approximated with the developed schemes.
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