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Special Section Articles

Higher-Order Linear-Time Unconditionally Stable Alternating Direction Implicit Methods for Nonlinear Convection-Diffusion Partial Differential Equation Systems

[+] Author and Article Information
Oscar P. Bruno

Computing and Mathematical Sciences,
California Institute of Technology,
e-mail: obruno@caltech.edu

Edwin Jimenez

Computing and Mathematical Sciences,
California Institute of Technology,
e-mail: jimenez@caltech.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 2, 2013; final manuscript received January 29, 2014; published online April 28, 2014. Assoc. Editor: Ye Zhou.

J. Fluids Eng 136(6), 060904 (Apr 28, 2014) (7 pages) Paper No: FE-13-1695; doi: 10.1115/1.4026868 History: Received December 02, 2013; Revised January 29, 2014

Abstract

We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDFs) of order $p≥2$, for the numerical solution of two-dimensional, time-dependent, nonlinear, convection-diffusion partial differential equation (PDE) systems in Cartesian domains. The proposed algorithms, which do not require the solution of nonlinear systems, additionally produce solutions of spectral accuracy in space through the use of Chebyshev approximations. In particular, these methods give rise to minimal artificial dispersion and diffusion and they therefore enable use of relatively coarse discretizations to meet a prescribed error tolerance for a given problem. A variety of numerical results presented in this text demonstrate high-order accuracy and, for the particular cases of $p=2,3$, unconditional stability.

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References

Beam, R. M. and Warming, R. F., 1978, “An Implicit Factored Scheme for the Compressible Navier-Stokes Equations,” AIAA J., 16(4), pp. 393–402.
Witelski, T. P. and Bowen, M., 2003, “ADI Schemes for Higher-Order Nonlinear Diffusion Equations,” Appl. Numer. Math., 45(2), pp. 331–351.
Bruno, O. P. and Lyon, M., 2010, “High-Order Unconditionally Stable FC-AD Solvers for General Smooth Domains I. Basic Elements,” J. Comput. Phys., 229, pp. 2009–2033.
Lyon, M. and Bruno, O. P., 2010, “High-Order Unconditionally Stable FC-AD Solvers for General Smooth Domains II. Elliptic, Parabolic and Hyperbolic PDEs; Theoretical Considerations,” J. Comput. Phys., 229, pp. 3358–3381.
Albin, N. and Bruno, O. P., 2011, “A Spectral FC Solver for the Compressible Navier–Stokes Equations in General Domains I: Explicit Time-Stepping,” J. Comput. Phys., 230(16), pp. 6248–6270.
Quarteroni, A., Sacco, R., and Saleri, F., 2000, Numerical Mathematics (Texts in Applied Mathematics), Springer, Paris.
Lee, J. and Fornberg, B., 2004, “Some Unconditionally Stable Time Stepping Methods for the 3D Maxwell's Equations,” J. Comput. Appl. Math., 166(2), pp. 497–523.

Figures

Fig. 2

Temporal convergence as Δt→0, using various spatial resolutions (Nx=Ny=N), of the approximate solution to the system (3) over [0,1]2. Maximum errors versus the time step Δt are obtained by means of various methods. (a) second-order spatial finite differences with N=50,100,200 and BDF(2)-ADI. (b) Chebyshev spatial approximation with N=20 and BDF(2)-ADI. (c) Chebyshev spatial approximation with N=20 and BDF(3)-ADI.

Fig. 3

Stability of the BDF(2)-ADI and BDF(3)-ADI temporal schemes when used in conjunction with the Chebyshev spatial approximations, as demonstrated by the display of the maximum error at a final time T=1000 with a fixed spatial resolution N=20 and various time-steps. (a) Δt=1. (b) Δt=10-1. (c) Δt=10-2.

Fig. 1

Burgers system solution in a Cartesian domain using BDF(3)-ADI and Chebyshev approximations

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