메뉴 건너뛰기




Volumn , Issue , 2005, Pages 1-279

Numerical solution of partial differential equations: An introduction

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84924514108     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9780511812248     Document Type: Book
Times cited : (864)

References (81)
  • 4
    • 84966233568 scopus 로고
    • Multilevel adaptive solutions to boundary-value problems
    • Brandt, A. (1977). Multilevel adaptive solutions to boundary-value problems. Math. Comp. 31, 333–90.
    • (1977) Math. Comp , vol.31 , pp. 333-390
    • Brandt, A.1
  • 6
    • 0042137401 scopus 로고    scopus 로고
    • Multi-symplectic structure and wave propagation
    • Bridges, T. J. (1997). Multi-symplectic structure and wave propagation. Math. Proc. Camb. Philos. Soc. 121, 147–90.
    • (1997) Math. Proc. Camb. Philos. Soc , vol.121 , pp. 147-190
    • Bridges, T.J.1
  • 8
    • 0346235067 scopus 로고
    • A necessary and sufficient condition for stability of difference schemes for initial value problems
    • Buchanan, M. L. (1963a). A necessary and sufficient condition for stability of difference schemes for initial value problems. J. Soc.Indust. Appl. Math. 11, 919–35.
    • (1963) J. Soc , vol.11 , pp. 919-935
    • Buchanan, M.L.1
  • 9
    • 0000266155 scopus 로고
    • A necessary and sufficient condition for stability of difference schemes for second order initial value problems
    • Buchanan, M. L. (1963b). A necessary and sufficient condition for stability of difference schemes for second order initial value problems. J. Soc.Indust. Appl. Math. 11, 474–501.
    • (1963) J. Soc , vol.11 , pp. 474-501
    • Buchanan, M.L.1
  • 12
    • 10444239968 scopus 로고    scopus 로고
    • Continuous dependence and error estimation forviscosity methods
    • Cockburn, B. (2003). Continuous dependence and error estimation forviscosity methods. Acta Numerica 12, 127–80.
    • (2003) Acta Numerica , vol.12 , pp. 127-180
    • Cockburn, B.1
  • 13
    • 34547419627 scopus 로고
    • The piecewise parabolic method (Ppm) for gas-dynamical simulations
    • Colella, P. and Woodward, P.R. (1984). The piecewise parabolic method (ppm) for gas-dynamical simulations. J. of Comput.Phys. 54, 174–201.
    • (1984) J. of Comput , vol.54 , pp. 174-201
    • Colella, P.1    Woodward, P.R.2
  • 16
    • 0003908527 scopus 로고
    • Über die partiellen differenzengleichungen der mathematischen physik
    • Courant, R., Friedrichs, K.O. and Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Math.Ann. 100, 32–74.
    • (1928) Math , vol.100 , pp. 32-74
    • Courant, R.1    Friedrichs, K.O.2    Lewy, H.3
  • 19
    • 84953600891 scopus 로고
    • A practical method for numerical evaluation of solutions of partial differential equations of the heatconduction type
    • Crank, J. and Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heatconduction type. Proc. Camb. Philos. Soc. 43, 50–67.
    • (1947) Proc. Camb. Philos. Soc , vol.43 , pp. 50-67
    • Crank, J.1    Nicolson, P.2
  • 20
    • 84967782959 scopus 로고
    • On the numerical solution of the heat conduction problems in two and three variables
    • Douglas, J. Jr. and Rachford, H. H. Jr. (1956). On the numerical solution of the heat conduction problems in two and three variables. Trans. Amer. Math. Soc. 82, 421–39.
    • (1956) Trans. Amer. Math. Soc , vol.82 , pp. 421-439
    • Douglas, J.1    Rachford, H.H.2
  • 22
    • 84924504940 scopus 로고    scopus 로고
    • Dyakonov, E.G. (1964). Difference schemes of second order accuracy with a splitting operator for parabolic equations without mixed partial derivatives. Zh. Vychisl. Mat. i Mat. Fiz., Moscow 4, 935–41.
    • D1    Dyakonov, E.G.2
  • 24
    • 84966208277 scopus 로고
    • One-sided difference approximations for nonlinear conservation laws
    • Engquist, B. and Osher, O. (1981). One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36, 321–52.
    • (1981) Math. Comp , vol.36 , pp. 321-352
    • Engquist, B.1    Osher, O.2
  • 25
    • 0003448545 scopus 로고    scopus 로고
    • Graduate Studies in Mathematics. Providence, Rhode Island, American Mathematical Society
    • Evans, L.C. (1998). Partial Differential Equations, Graduate Studies in Mathematics. Providence, Rhode Island, American Mathematical Society.
    • (1998) Partial Differential Equations
    • Evans, L.C.1
  • 26
    • 0000935266 scopus 로고
    • A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics
    • Godunov, S. K. (1959). A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–306.
    • (1959) Mat. Sb , vol.47 , pp. 271-306
    • Godunov, S.K.1
  • 28
    • 0004236492 scopus 로고    scopus 로고
    • 3rd edn. Baltimore, Johns Hopkins University Press
    • Golub, G.H. and Van Loan, C. F. (1996). Matrix Computations, 3rd edn. Baltimore, Johns Hopkins University Press.
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 29
    • 0035273564 scopus 로고    scopus 로고
    • Strong stabilitypreserving high-order time discretisation methods
    • Gottlieb, S., Shu, C.W. and Tadmor, E. (2001). Strong stabilitypreserving high-order time discretisation methods. SIAM Rev. 43, 89–112.
    • (2001) SIAM Rev , vol.43 , pp. 89-112
    • Gottlieb, S.1    Shu, C.W.2    Tadmor, E.3
  • 30
    • 84968486120 scopus 로고
    • Stability theory of difference approximations for mixed initial boundary value problems ii
    • Gustaffson, B., Kreiss, H.O. and Sundström, A. (1972). Stability theory of difference approximations for mixed initial boundary value problems ii. Math. Comp. 24(119), 649–86.
    • (1972) Math. Comp , vol.24 , Issue.119 , pp. 649-686
    • Gustaffson, B.1    Kreiss, H.O.2    Sundström, A.3
  • 34
    • 0021381876 scopus 로고
    • On a class of high resolution total-variation-stable finite-difference schemes
    • Harten, A. (1984). On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21, 1–23.
    • (1984) SIAM J. Numer. Anal , vol.21 , pp. 1-23
    • Harten, A.1
  • 35
    • 0000135303 scopus 로고
    • Methods of conjugate gradients for solving linear systems
    • Hestenes, M.R. and Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–36.
    • (1952) J. Res. Nat. Bur. Stand , vol.49 , pp. 409-436
    • Hestenes, M.R.1    Stiefel, E.2
  • 36
    • 84980083129 scopus 로고
    • On the integration of parabolic equations by difference methods
    • John, F. (1952). On the integration of parabolic equations by difference methods. Comm. Pure Appl. Math. 5, 155.
    • (1952) Comm. Pure Appl. Math , vol.5 , pp. 155
    • John, F.1
  • 37
    • 0000958123 scopus 로고
    • A new finite difference scheme for parabolic problems
    • B. Hubbard (ed.), Academic Press
    • Keller, H.B. (1971). A new finite difference scheme for parabolic problems. In B. Hubbard (ed.), Numerical Solution of Partial DifferentialEquations II, SYNSPADE 1970, Academic Press, pp. 327–50.
    • (1971) Numerical Solution of Partial Differential , pp. 327-350
    • Keller, H.B.1
  • 38
    • 38249011463 scopus 로고
    • Maximumnormcontractivity of discretization schemes for the heat equation
    • Kraaijevanger, J. F. B.M. (1992). Maximumnormcontractivity of discretization schemes for the heat equation. Appl. Numer. Math. 99, 475–92.
    • (1992) Appl. Numer. Math , vol.99 , pp. 475-492
    • Kraaijevanger, J.1
  • 39
    • 0001531681 scopus 로고
    • Über die stabilitätsdefinition für differenzengleichungen die partielle differentialgleichungen approximieren
    • Kreiss, H.O. (1962). Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren. Nordisk Tidskr. Informations-Behandling 2, 153–81.
    • (1962) Nordisk Tidskr. Informations-Behandling , vol.2 , pp. 153-181
    • Kreiss, H.O.1
  • 40
    • 84980076945 scopus 로고
    • On difference approximations of the dissipative type for hyperbolic differential equations
    • Kreiss, H.O. (1964). On difference approximations of the dissipative type for hyperbolic differential equations. Comm. Pure Appl. Math. 17, 335–53.
    • (1964) Comm. Pure Appl. Math , vol.17 , pp. 335-353
    • Kreiss, H.O.1
  • 42
    • 84980086027 scopus 로고
    • Survey of the stability of linear finite difference equations
    • Lax, P. D. and Richtmyer, R. D. (1956). Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267–93.
    • (1956) Comm. Pure Appl. Math , vol.9 , pp. 267-293
    • Lax, P.D.1    Richtmyer, R.D.2
  • 45
    • 0003671760 scopus 로고
    • Lectures in Mathematics ETH Zurich, 2nd edn. Basel, Birkhauser, Verlag
    • LeVeque, R. J. (1992). Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zurich, 2nd edn. Basel, BirkhauserVerlag.
    • (1992) Numerical Methods for Conservation Laws
    • LeVeque, R.J.1
  • 47
    • 0040045610 scopus 로고
    • Fourier analysis of the sor iteration
    • LeVeque, R. J. and Trefethen, L. N. (1988). Fourier analysis of the SOR iteration. IMA J. Numer. Anal. 8 (3), 273–9.
    • (1988) IMA J. Numer. Anal , vol.8 , Issue.3 , pp. 273-279
    • LeVeque, R.J.1    Trefethen, L.N.2
  • 48
    • 0032014411 scopus 로고    scopus 로고
    • From semi-discrete to fully discrete: Stability of runge–kutta schemes by the energy method
    • Levy, D. and Tadmor, E. (1998). From semi-discrete to fully discrete: stability of Runge–Kutta schemes by the energy method. SIAMRev. 40, 40–73.
    • (1998) SIAM , vol.40 , pp. 40-73
    • Levy, D.1    Tadmor, E.2
  • 49
    • 84924504938 scopus 로고    scopus 로고
    • Lighthill, M.J. (1978). Waves in Fluids. Cambridge, Cambridge University Press.
    • Lighthill, M.J.1
  • 50
    • 0002262211 scopus 로고
    • Stability and convergence in numerical analysis iii: Linear investigation of nonlinear stability
    • López-Marcos, J. C. and Sanz-Serna, J. M. (1988). Stability and convergence in numerical analysis iii: linear investigation of nonlinear stability. IMA J. Numer. Anal. 8 (1), 71–84.
    • (1988) IMA J. Numer. Anal , vol.8 , Issue.1 , pp. 71-84
    • López-Marcos, J.C.1    Sanz-Serna, J.M.2
  • 51
    • 24944577283 scopus 로고
    • False reflection of waves at the boundary due to the use of finite differences
    • Matsuno, T. (1966). False reflection of waves at the boundary due to the use of finite differences. J. Meteorol. Soc. Japan 44 (2), 145–57.
    • (1966) J. Meteorol. Soc. Japan , vol.44 , Issue.2 , pp. 145-157
    • Matsuno, T.1
  • 52
    • 0000393697 scopus 로고
    • On the location of zeros of certain classes of polynomials with applications to numerical analysis
    • Miller, J. (1971). On the location of zeros of certain classes of polynomials with applications to numerical analysis. J. Inst. Math. Appl. 8, 397–406.
    • (1971) J. Inst. Math. Appl , vol.8 , pp. 397-406
    • Miller, J.1
  • 55
    • 0002058827 scopus 로고
    • The numerical solution of parabolic and elliptic differential equations
    • Peaceman, D. W. and Rachford, H. H. Jr. (1955). The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl.Math. 3, 28–41.
    • (1955) J. Soc. Indust. Appl , vol.3 , pp. 28-41
    • Peaceman, D.W.1    Rachford, H.H.2
  • 56
    • 84924504936 scopus 로고
    • paper presented at the First Congress of the French Association for Computation, Grenoble, France
    • Preissmann, A. (1961). Propagation des intumescences dans les canaux et rivières, paper presented at the First Congress of the French Association for Computation, Grenoble, France.
    • (1961) Propagation Des Intumescences Dans Les Canaux Et rivières
    • Preissmann, A.1
  • 58
    • 0001444647 scopus 로고    scopus 로고
    • Backward error analysis for numerical integration
    • Reich, S. (1999). Backward error analysis for numerical integration. SIAM J. Numer. Anal. 36, 1549–1570.
    • (1999) SIAM J. Numer. Anal , vol.36 , pp. 1549-1570
    • Reich, S.1
  • 60
    • 2942757053 scopus 로고
    • Approximateriemannsolvers, parametervectors, and difference schemes
    • Roe, P. L. (1981).ApproximateRiemannsolvers, parametervectors, and difference schemes. J. of Comput. Phys. 43, 357–72.
    • (1981) J. of Comput. Phys , vol.43 , pp. 357-372
    • Roe, P.L.1
  • 62
    • 0000048673 scopus 로고
    • Ageneralised minimal residual algorithm for solving nonsymmetric linear systems
    • Saad, Y. and Schultz, M. H. (1986).Ageneralised minimal residual algorithm for solving nonsymmetric linear systems. J. Sci. Stat. Comput.7, 856–69.
    • (1986) J. Sci. Stat. Comput , vol.7 , pp. 856-869
    • Saad, Y.1    Schultz, M.H.2
  • 65
    • 27144497812 scopus 로고
    • Wiener–hopf difference equations
    • Strang, G. (1964). Wiener–Hopf difference equations. J. Math. Mech. 13(1), 85–96.
    • (1964) J. Math. Mech , vol.13 , Issue.1 , pp. 85-96
    • Strang, G.1
  • 68
    • 24644493466 scopus 로고    scopus 로고
    • Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems
    • Tadmor, E. (2003). Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12, 451–512.
    • (2003) Acta Numerica , vol.12 , pp. 451-512
    • Tadmor, E.1
  • 69
    • 5544288869 scopus 로고
    • A stable difference scheme for the mixed boundary value problem for a hyperbolic first order system in two dimensions
    • Thomée, V. (1962). A stable difference scheme for the mixed boundary value problem for a hyperbolic first order system in two dimensions. J. Soc. Indust. Appl. Math. 10, 229–45.
    • (1962) J. Soc. Indust. Appl. Math , vol.10 , pp. 229-245
    • Thomée, V.1
  • 70
    • 0000125555 scopus 로고
    • Group velocity in finite difference schemes
    • Trefethen, L. N. (1982). Group velocity in finite difference schemes. SIAM Rev. 24, 113–36.
    • (1982) SIAM Rev , vol.24 , pp. 113-136
    • Trefethen, L.N.1
  • 72
    • 34249988639 scopus 로고
    • Towards the ultimate conservative difference scheme. Ii: Monotonicity and conservation combined in a second order scheme
    • van Leer, B. (1974). Towards the ultimate conservative difference scheme. II: monotonicity and conservation combined in a second order scheme. J. of Comput. Phys. 14, 361–70.
    • (1974) J. of Comput. Phys , vol.14 , pp. 361-370
    • Van Leer, B.1
  • 73
    • 2442433925 scopus 로고
    • Towards the ultimate conservative difference scheme v. A second order sequel to godunov’s method
    • van Leer, B. (1979). Towards the ultimate conservative difference scheme V. a second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–36.
    • (1979) J. Comput. Phys , vol.32 , pp. 101-136
    • Van Leer, B.1
  • 75
    • 49549156305 scopus 로고
    • Themodified equation approach to the stability and accuracy of finite difference methods
    • Warming, R. F. and Hyett, B. J. (1974).Themodified equation approach to the stability and accuracy of finite difference methods. J. of Comput.Phys. 14, 159–79.
    • (1974) J. of Comput , vol.14 , pp. 159-179
    • Warming, R.F.1    Hyett, B.J.2
  • 79
    • 0348090281 scopus 로고
    • First differential approximation method and approximate viscosity of difference schemes
    • II
    • Yanenko, N. N. and Shokin, Y. I. (1969). First differential approximation method and approximate viscosity of difference schemes. Phys. OfFluids 12, Suppl. II, 28–33.
    • (1969) Phys. Of , vol.12 , pp. 28-33
    • Yanenko, N.N.1    Shokin, Y.I.2
  • 81
    • 0034640067 scopus 로고    scopus 로고
    • Multisymplectic geometry andmultisymplectic preissmann scheme for the kdv equation
    • Zhao, P. F. and Quin, M. Z. (2000). Multisymplectic geometry andmultisymplectic Preissmann scheme for the kdv equation. J. Phys.A33, 3613–26.
    • (2000) J. Phys , vol.A33 , pp. 3613-3626
    • Zhao, P.F.1    Quin, M.Z.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.