Analysis of a compact multi-step ADI method for linear parabolic equation

References

• Bialeck B, Ganesh M, Mustaphs K. An ADI Petrov-Galerkin method with quadrature for parabolic problems. Numer Methods Partial Differ Equ. 2009;25:1129–1148.
   Web of Science ®Google Scholar
• Wang T. Alternating direction finite volume element methods for 2D parabolic partial differential equation. Numer Methods Partial Differ Equ. 2008;24:24–40.
   Web of Science ®Google Scholar
• Bialeck B, Fernandes RI. An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM Numer Anal. 1999;36:144–1434.
   Web of Science ®Google Scholar
• Zhao J, Dai W, Zhang Y. Fourth-order compact schemes for solving multidimensional heat problems with Neumann Boundary conditions. Numer Methods Partial Differ Equ. 2008;24:165–178.
   Web of Science ®Google Scholar
• Dai W. Compact ADI method for solving parabolic differential equations, Numer Methods Partial Differ Equ. 2002;18:129–142.
   Web of Science ®Google Scholar
• Wang Y, Zhang H. Error and extrapolation of a compact LOD method for parabolic differential equations. J Comput Appl Math. 2011;235:1367–1382.
   Web of Science ®Google Scholar
• Liao H, Sun Z. Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer Methods Partial Differ Equ. 2010;26:37–60.
   Web of Science ®Google Scholar
• Li J, Chen Y, Liu G. High-order compact ADI methods for parabolic equations. Comput Math Appl. 2006;52:1343–1356.
   Web of Science ®Google Scholar
• Sun Z LY. A compact alternate direct implicit difference method for the reaction diffusion equations. Math Numer Sini. 2005;20:209–224 (in Chinese).
   Google Scholar
• Wang QJ. T, A compact locally one-dimensional finite difference method for nonhomogeneous parabolic differential equations. Int J Numer Meth Biomed Engng. 2011;27:128–142.
   Web of Science ®Google Scholar
• Liao W, Zhu J, Khallq AQM. An efficient high-order algorithm for solving systems of reaction-diffusion equations. Numer Methods Partial Differ Equ. 2002;18:340–354.
   Web of Science ®Google Scholar
• Gu Y, Liao W, Zhu J. An efficient high-order algorithm for solving systems of 3-D reaction-diffusion equations. J Comput Appl Math. 2003;155:1–17.
   Web of Science ®Google Scholar
• Wang Y, Wang J. High-order compact ADI method with monotone iterative procedure systems reaction-diffusion equations. Comput Math Appl. 2011;233:2434–2451.
   Web of Science ®Google Scholar
• Peaceman WD. Rachford HH numerical solution parabolic elliptic differential equations J Soc Ind Appl Math. 1955;3:28–41.
   Google Scholar
• Dai W. A generalized Peaceman-Rachford ADI scheme for solving two-dimensional parabolic differential equations. J Sci Comput. 1997;12:353–360.
   Google Scholar
• Gourlay AR, Mitchell AR. The equivalence of certain alternating direction and locally one-dimensional difference methods. SIAM J Numer Anal. 1969;6:37–46.
   Web of Science ®Google Scholar
• Arraás A, Portero L, Jorge JC. Locally linearized fractional step methods for nonlinear parabolic problems. J Comput Appl Math. 2010;234:1117–1128.
   Web of Science ®Google Scholar
• Karaa S. Improved accuracy for the approximate factorization of parabolic equation. Comput. 2009;86:23–36.
   Web of Science ®Google Scholar
• Douglas JJ, Kim S. Improved accuracy for locally one-dimensional methods for parabolic equation. Math Models Methods Appl Sci. 2001;11:1563–1579.
   Web of Science ®Google Scholar
• Lim H, Kim S, Douglas JJ. Numerical methods for viscous and nonviscous wave equations. Appl Numer Math. 2007;57:194–212.
   Web of Science ®Google Scholar
• Lele RK. Compact finite difference schemes with spectral-like resolution. J Comput Phys. 1992;103:16–42.
   Web of Science ®Google Scholar
• Zhang Q, Ran M, Li Z. Numerical approximation for two-dimensional neutral parabolic differential equations with delay. Inter J Model Simul. 2016;36(1–2):12–19.
   Google Scholar
• Zhang Q, Zhang C. A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations. Appl Math Lett. 2013;26(2):306–312.
   Web of Science ®Google Scholar
• Xie J, Deng D, Zheng H. Fourth-order difference solvers for nonlinear delayed fractional sub-diffusion equations with variable coefficients. Inter J Model Simul. 2017;37(2):241–251.
   Google Scholar
• Tangman DY, Gopaul A, Bhuruth M. A fast high-order finite difference algorithm for pricing American options. J Comput Appl Math. 2008;22:17–29.
   Web of Science ®Google Scholar
• Deng D, Zhang Z. A new high-order algorithm for a class of nonlinear evolution equation. J Phys A Math Theor. 2008;41(015202):(17.
   Google Scholar
• Deng D, Zhang C. A new fourth-order numerical algorithm for a class of three-dimensional nonlinear evolution equations. Numer Methods Partial Differ Equ. 2013;29:102–130.
   Web of Science ®Google Scholar
• Karaa S. A hybrid Padé ADI scheme of high-order for convection-diffusion problems. Int J Numer Meth Fluids. 2010;64:532–548.
   Web of Science ®Google Scholar
• Hairer E, Wanner G. Solving ordinary differential equations II: stiff and differential-algebraic problems (second edition). Beijing: Science Press; 2006.
   Google Scholar
• Hundsdorfer W. Numerical solution of advection-diffusion-reaction equations, lecture notes. Thomas Stieltjes Institute; 2000. Available from: http://homepages.cwi.nl/willem/Coll_AdvDiffReac/notes.pdf
   Google Scholar
• Beam M, Warming RF. An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. J Comput Phys. 1976;22:87–110.
   Web of Science ®Google Scholar
• Zhang Z. The multistep finite difference fractional steps method for a class of viscous wave equations. Math Meth Appl Sci. 2011;34:442–454.
   Web of Science ®Google Scholar
• Bialecki B, Fernandes RI. An alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations. SIAM J Numer Anal. 2009;47:3429–3450.
   Web of Science ®Google Scholar
• Ikonen S, Toivanen J. Operator splitting methods for pricing American options under stochastic volatility. Numer Math. 2009;113:299–324.
   Web of Science ®Google Scholar