Abstract
A collection of Matlab routines that compute derivative approximations of arbitrary functions using high-order compact finite difference schemes is presented. Tenth-order accurate compact finite difference schemes for first and second derivative approximations and sixth-order accurate compact finite difference schemes for third and fourth derivative approximations are discussed for the functions with periodic boundary conditions. Fourier analysis of compact finite difference schemes is explained, and it is observed that compact finite difference schemes have better resolution characteristics when compared to classical finite difference schemes. Compact finite difference schemes for the functions with Dirichlet and Neumann boundary conditions are also discussed. Moreover, compact finite difference schemes for partial derivative approximations of functions in two variables are also given. For each case a Matlab routine is provided to compute the differentiation matrix and results are validated using the test functions.
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- Y. Adam. 1975. A hermitian finite difference method for the solution of parabolic equations. Comp. Maths. Appls. 1, 393--406 Google ScholarCross Ref
- Y. Adam. 1977. Highly accurate compact implicit methods and boundary conditions. J. Comput. Phys. 24, 10--22. Google ScholarCross Ref
- C. Cheong and S. Lee. 2001. Grid-optimized dispersion-relation-preserving schemes on general geometries for computational aeroacoustics. J. Comput. Phys. 174, 248--276. Google ScholarDigital Library
- P. C. Chu and C. Fan. 1998. A three-point combined compact difference scheme. J. Comput. Phys. 140, 370--399. Google ScholarDigital Library
- P. C. Chu and C. Fan. 1999. A three-point sixth-order nonuniform combined compact difference scheme. J. Comput. Phys. 148, 663--674. Google ScholarDigital Library
- B. During and M. Fournie. 2012. High-order compact finite difference scheme for option pricing in stochastic volatility models. J. Comput. Appl. Math. 236, 4462--4473. Google ScholarDigital Library
- B. During, M. Fournie, and A. Jungel. 2004. Convergence of high-order compact finite difference scheme for a nonlinear Black-Scholes equation. Math. Model. Numer. Anal. 38, 359--369. Google ScholarCross Ref
- L. Gamet, F. Ducros, F. Nicoud, and T. Poinsot. 1999. Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulation of compressible flows. Int. J. Numer. Meth. Fluids 29, 159--191. Google ScholarCross Ref
- L. Ge and J. Zhang. 2001. High accuracy solution of convection diffusion equation with boundary layers on nonuniform grids. J. Comput. Phys. 171, 560--571. Google ScholarDigital Library
- W. J. Geodheer and J. H. H. M. Potters. 1985. A compact finite difference scheme on a non-equidistant mesh. J. Comput. Phys. 61, 269--279. Google ScholarCross Ref
- R. S. Hirsh. 1975. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90--109. Google ScholarCross Ref
- V. Kumar. 2009. High-order compact finite-difference scheme for singularly-perturbed reaction-diffusion problems on a new mesh of Shishkin type. J. Optim. Theory. Appl. 143, 123--147. Google ScholarDigital Library
- S. K. Lele. 1992. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16--42. Google ScholarCross Ref
- K. Mahesh. 1998. A family of high order finite difference schemes with good spectral resolution. J. Comput. Phys. 145, 332--358. Google ScholarDigital Library
- H. L. Meitz and H. F. Fasel. 2000. A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation. J. Comput. Phys. 157, 371--403. Google ScholarDigital Library
- Shuvam Sen. 2013. A new family of (5,5)CC-4OC schemes applicable for unsteady Navier-Stokes equations. J. Comput. Phys. 251, 251--271. Google ScholarDigital Library
- Shuvam Sen. 2016. Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives. Comput. Fluids 134--135, 81--89.Google Scholar
- T. K. Sengupta, G. Ganeriwal, and S. De. 2003. Analysis of central and upwind compact schemes. J. Comput. Phys. 192, 677--694. Google ScholarDigital Library
- J. S. Shang. 1999. High-order compact-difference schemes for time-dependent maxwell equations. J. Comput. Phys. 153, 312--333. Google ScholarDigital Library
- R. K. Shukla, M. Tatineni, and X. Zhong. 2007. Very high-order compact finite difference schemes for non-uniform grid for incompressible Navier-Stokes equations. J. Comput. Phys. 224, 1064--1094. Google ScholarDigital Library
- R. K. Shukla and X. Zhong. 2005. Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation. J. Comput. Phys. 204, 404--429. Google ScholarDigital Library
- E. Weinan and J. G. Liu. 1996. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122--138. Google ScholarDigital Library
- J. Zhao, W. Dai, and T. Niu. 2007. Fourth order compact scheme of a heat conduction problem with Neumann boundary conditions. Numer. Methods Partial Differential Eq. 12, 949--959. Google ScholarCross Ref
Index Terms
Algorithm 986: A Suite of Compact Finite Difference Schemes
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