Improved ADI scheme for linear hyperbolic equations: extension to nonlinear cases and compact ADI schemes. (English) Zbl 1377.65113
Summary: By introducing appropriate intermediate variables and simplifying inhomogeneous terms of the traditional alternating direction implicit (ADI) schemes, an improved ADI scheme is proposed for linear hyperbolic equations and this ADI scheme is extended to nonlinear hyperbolic equations and compact ADI schemes in the present work. Meanwhile, the boundary and initial conditions are carefully analyzed to match the accuracy of these improved ADI schemes. Although the common (without compactification) and compact improved ADI schemes have the same accuracy with the corresponding traditional ADI schemes, respectively, it is not just a simple variant of ADI schemes for hyperbolic equations. Both theoretical analysis and numerical experiments show that compared with the traditional ADI scheme, the improved ADI scheme is more efficient for linear hyperbolic equations and more stable for nonlinear hyperbolic equations without loss of accuracy.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
hyperbolic equations; boundary condition; numerical efficiency; numerical stability; finite difference method; alternating direction implicit schemes; linear; nonlinear; numerical experimentReferences:
| [1] | Greiner, W.: Classical electrodynamics. Springer, Berlin (2012) |
| [2] | Mullis, A.M.: Rapid solidification within the framework of a hyperbolic conduction model. Int. J. Heat Mass Trans. 40(17), 4085-4094 (1997) · Zbl 0921.76177 · doi:10.1016/S0017-9310(97)00062-8 |
| [3] | Galenko, P.K., Danilov, D.A.: Hyperbolic self-consistent problem of heat transfer in rapid solidification of supercooled liquid. Phys. Lett. A 278(3), 129-138 (2000) · doi:10.1016/S0375-9601(00)00772-6 |
| [4] | Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32-74 (1928) · JFM 54.0486.01 · doi:10.1007/BF01448839 |
| [5] | Peaceman, D.W., Rachford, H.H., Henry, H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28-41 (1955) · Zbl 0067.35801 · doi:10.1137/0103003 |
| [6] | Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421-439 (1956) · Zbl 0070.35401 · doi:10.1090/S0002-9947-1956-0084194-4 |
| [7] | Gourlay, A., Mitchell, A.R.: The equivalence of certain alternating direction and locally one-dimensional difference methods. SIAM J. Numer. Anal. 6, 37-46 (1969) · Zbl 0175.16202 · doi:10.1137/0706004 |
| [8] | Lees, M.: Alternating direction methods for hyperbolic differential equations. J. Soc. Ind. Appl. Math. 10, 610-616 (1962) · Zbl 0111.29204 · doi:10.1137/0110046 |
| [9] | Gourlay, A.R., Mitchell, A.R.: A classification of split difference methods for hyperbolic equations in several space dimensions. SIAM J. Numer. Anal. 6, 62-71 (1969) · Zbl 0175.16203 · doi:10.1137/0706006 |
| [10] | Mohanty, R.K.: An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions. Appl. Math. Comput. 152, 799-806 (2004) · Zbl 1077.65093 |
| [11] | Mohanty, R.K.: An operator splitting technique for an unconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable coefficients. Appl. Math. Comput. 162, 549-557 (2005) · Zbl 1063.65084 |
| [12] | Mohanty, R.K., Jain, M.K., Arora, U.: An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions. Int. J. Comput. Math. 79, 133-142 (2002) · Zbl 0995.65093 · doi:10.1080/00207160211918 |
| [13] | Ciment, M., Leventhal, S.H.: Higher order compact implicit schemes for the wave equation. Math. Comput. 29, 985-994 (1975) · Zbl 0309.35043 · doi:10.1090/S0025-5718-1975-0416049-2 |
| [14] | Dai, W., Nassar, R.: Compact ADI method for solving parabolic differential equations. Numer. Meth. Part. Differ. Equ. 18, 129-142 (2002) · Zbl 1004.65086 · doi:10.1002/num.1037 |
| [15] | Ding, H., Zhang, Y.: A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation. J. Comput. Appl. Math. 230, 626-632 (2009) · Zbl 1168.65373 · doi:10.1016/j.cam.2009.01.001 |
| [16] | Mohanty, R.K., Jain, M.K.: An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation. Numer. Meth. Part. Differ. Equ. 17, 684-688 (2001) · Zbl 0990.65101 · doi:10.1002/num.1034 |
| [17] | Dehghan, M., Shokri, A.: A numerical method for solving the hyperbolic telegraph equation. Numer. Meth. Part. Differ. Equ. 24, 1080-1093 (2008) · Zbl 1145.65078 · doi:10.1002/num.20306 |
| [18] | Mohanty, R.K.: New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math. 86, 2061-2071 (2009) · Zbl 1181.65112 · doi:10.1080/00207160801965271 |
| [19] | Deng, D., Zhang, C.: Application of a fourth-order compact ADI method to solve a two-dimensional linear hyperbolic equation. Int. J. Comput. Math. 90, 273-291 (2013) · Zbl 1278.65122 · doi:10.1080/00207160.2012.713475 |
| [20] | Deng, D., Zhang, C.: Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables. Numer. Alg. 63, 1-26 (2013) · Zbl 1269.65078 · doi:10.1007/s11075-012-9604-8 |
| [21] | Mohanty, R., Jain, M., Arora, U.: An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions. Int. J. Comput. Math. 79, 133-142 (2002) · Zbl 0995.65093 · doi:10.1080/00207160211918 |
| [22] | Karaa, S.: Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations. Int. J. Comput. Math. 87, 3030-3038 (2010) · Zbl 1210.65161 · doi:10.1080/00207160902878548 |
| [23] | Cui, M.: High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation. J. Comput. Appl. Math. 235, 837-849 (2010) · Zbl 1208.65126 · doi:10.1016/j.cam.2010.07.016 |
| [24] | Deng, D., Zhang, C.: A new fourth-order numerical algorithm for a class of three-dimensional nonlinear evolution equations. Numer. Meth. Part. Differ. Equ. 29, 102-130 (2013) · Zbl 1255.65158 · doi:10.1002/num.21701 |
| [25] | Jain, M.K.: Numerical Solution of Differential Equations. Wiley, New Delhi (1984) · Zbl 0536.65004 |
| [26] | Mohanty, R.K., George, K., Jain, M.K.: High accuracy difference schemes for a class of singular three space dimensional hyperbolic equations. Int. J. Comput. Math. 56, 185-198 (1995) · Zbl 0845.65046 · doi:10.1080/00207169508804400 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
