Optimized Compact-Difference-Based Finite-Volume Schemes for Linear Wave Phenomena

High-order compact-difference-based finite-volume schemes are developed, analyzed, and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time-domain. The formulation combines the primitive function approach with five-point spatially sixth- and fourth-order methods. Optimization in the semi-discrete case is achieved by minimizing dispersion and isotropy error. The fully discrete scheme is examined by adopting the classical fourth-order Runge?Kutta technique. Stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance. The scheme is then formally extended to multiple dimensions. Consistent boundary conditions are presented for the reconstruction operator as well as flux specification. Several calculations in one and three dimensions confirm the properties of the method.

[1]  Exact Integrations of Polynomials and High-Order Quadratures for Finite-Volume Methods with Arbitrary Polyhedral Grids , 1997 .

[2]  Shlomo Ta'asan,et al.  Finite difference schemes for long-time integration , 1994 .

[3]  Datta V. Gaitonde,et al.  Characteristic-based, time-dependent Maxwell equation solvers on a general curvilinear frame , 1995 .

[4]  Two compact cell-vertex methods for computational electromagnetics , 1991 .

[5]  H. Kreiss,et al.  On the stability definition of difference approximations for the initial boundary value problem , 1993 .

[6]  D. Gottlieb,et al.  Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes , 1994 .

[7]  Harold L. Atkins,et al.  A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems , 1993 .

[8]  Yen Liu,et al.  Analysis of numerical approaches for acoustic equations , 1993 .

[9]  H. Kreiss,et al.  Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems. II , 1972 .

[10]  R. Hirsh,et al.  Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique , 1975 .

[11]  D. Fyfe,et al.  Economical evaluation of Runge-Kutta formulae , 1966 .

[12]  Wright-Patterson Afb,et al.  Investigations on the Properties of a Finite-Volume, Time-Domain Method for Computational Electromagnetics , 1995 .

[13]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[14]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[15]  Bertil Gustafsson,et al.  Fourth-order difference methods for hyperbolic IBVPs , 1995 .

[16]  RADIATION BOUNDARY CONDITIONS , 1994 .

[17]  B. P. Leonard.  Order of accuracy of QUICK and related convection-diffusion schemes , 1995 .

[18]  Y. Adam,et al.  Highly accurate compact implicit methods and boundary conditions , 1977 .

[19]  W. H. Enright.  Analysis of error control strategies for continuous Runge-Kutta methods , 1989 .

[20]  D. Gottlieb,et al.  The Stability of Numerical Boundary Treatments for Compact High-Order Finite-Difference Schemes , 1993 .

[21]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[22]  Vijaya Shankar,et al.  A CFD-based finite-volume procedure for computational electromagnetics - Interdisciplinary applications of CFD methods , 1989 .

[23]  Samuel D. Conte,et al.  Elementary Numerical Analysis: An Algorithmic Approach , 1975 .

[24]  J. Bowles,et al.  Fourier Analysis of Numerical Approximations of Hyperbolic Equations , 1987 .

[25]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[26]  Chi-Wang Shu,et al.  A comparison of two formulations for high-order accurate essentially non-oscillatory schemes , 1994 .

[27]  C. Tam,et al.  Dispersion-relation-preserving finite difference schemes for computational acoustics , 1993 .