TY - JOUR

T1 - A compact fourth order scheme for the Helmholtz equation in polar coordinates

AU - Britt, S.

AU - Tsynkov, S.

AU - Turkel, E.

N1 - Funding Information:
This work was partially supported by the United States—Israel Binational Science Foundation (BSF), grant number 2008094. Research of the first and second authors was also supported in part by the US Air Force, grant number FA9550-07-1-0170, and US NSF, grant number DMS-0509695.

PY - 2010/10

Y1 - 2010/10

N2 - 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.

AB - 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.

KW - Compact finite differences

KW - Exact artificial boundary condition (ABC)

KW - Helmholtz equation

KW - High order accuracy

KW - Polar coordinates

KW - Variable coefficients

UR - http://www.scopus.com/inward/record.url?scp=77956189182&partnerID=8YFLogxK

U2 - 10.1007/s10915-010-9348-3

DO - 10.1007/s10915-010-9348-3

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AN - SCOPUS:77956189182

SN - 0885-7474

VL - 45

SP - 26

EP - 47

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 1-3

ER -