Computational Methods for Physicists pp 467-517 | Cite as

# Difference Methods for One-Dimensional PDE

## Abstract

Finite-difference methods for one-dimensional partial differential equations are introduced by first identifying the classes of equations upon which suitable discretizations are constructed. It is shown how parabolic equations and the corresponding boundary conditions are discretized such that a desired local order of error is achieved and that the discretization is consistent and yields a stable and convergent solution scheme. Convergence criteria are established for a variety of explicit and implicit difference schemes. Energy estimates and theorems on maxima are given as auxiliary tools that allow us to ascertain that the solutions are physically meaningful. Difference schemes for hyperbolic equations are introduced from the standpoint of the Courant–Friedrich–Lewy criterion, dispersion and dissipation. Various techniques for non-linear equations and equations of mixed type are given, including high-resolution schemes for equations that can be expressed in terms of conservation laws. The Problems include the (parabolic) diffusion and (hyperbolic) advection equation, Burgers equation, the shock-tube problem, Korteweg–de Vries equation, and the non-stationary linear and cubic Schrödinger equations.

## Keywords

Difference Scheme Implicit Scheme Partial Differential Equation Nicolson Scheme Homogeneous Dirichlet Condition## References

- 1.J.W. Thomas,
*Numerical Partial Differential Equations: Finite Difference Methods*. Springer Texts in Applied Mathematics, vol. 22 (Springer, Berlin, 1998) Google Scholar - 2.D. Knoll, J. Morel, L. Margolin, M. Shashkov, Physically motivated discretization methods. Los Alamos Sci.
**29**, 188 (2005) Google Scholar - 3.A. Tveito, R. Winther,
*Introduction to Partial Differential Equations*. Springer Texts in Applied Mathematics, vol. 29 (Springer, Berlin, 2005) zbMATHGoogle Scholar - 4.G.D. Smith,
*Numerical Solution of Partial Differential Equations*(Oxford University Press, Oxford, 2003) Google Scholar - 5.J.W. Thomas,
*Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations*. Springer Texts in Applied Mathematics, vol. 33 (Springer, Berlin, 1999) zbMATHGoogle Scholar - 6.E. Godlewski, P.-A. Raviart,
*Numerical Approximations of Hyperbolic Systems of Conservation Laws*(Springer, Berlin, 1996) Google Scholar - 7.R.J. LeVeque,
*Numerical Methods for Conservation Laws*(Birkhäuser, Basel, 1990) zbMATHCrossRefGoogle Scholar - 8.P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal.
**21**, 995 (1984) MathSciNetADSzbMATHCrossRefGoogle Scholar - 9.D. Eisen, On the numerical analysis of a fourth order wave equation. SIAM J. Numer. Anal.
**4**, 457 (1967) MathSciNetADSzbMATHCrossRefGoogle Scholar - 10.D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave. Philos. Mag. Ser. 5
**39**, 422 (1895) zbMATHCrossRefGoogle Scholar - 11.B. Fornberg, G.B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. R. Soc. Lond. Ser. A
**289**, 373 (1978) MathSciNetADSzbMATHCrossRefGoogle Scholar - 12.X. Lai, Q. Cao, Some finite difference methods for a kind of GKdV equations. Commun. Numer. Methods Eng.
**23**, 179 (2007) MathSciNetzbMATHCrossRefGoogle Scholar - 13.Q. Cao, K. Djidjeli, W.G. Price, E.H. Twizell, Computational methods for some non-linear wave equations. J. Eng. Math.
**35**, 323 (1999) MathSciNetzbMATHCrossRefGoogle Scholar - 14.K. Kormann, S. Holmgren, O. Karlsson, Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. J. Chem. Phys.
**128**, 184101 (2008) ADSCrossRefGoogle Scholar - 15.C. Lubich, in
*Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms*, ed. by J. Grotendorst, D. Marx, A. Muramatsu. NIC Series, vol. 10 (John von Neumann Institute for Computing, Jülich, 2002), p. 459 Google Scholar - 16.W. van Dijk, F.M. Toyama, Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E
**75**, 036707 (2007) MathSciNetADSCrossRefGoogle Scholar - 17.T.N. Truong et al., A comparative study of time dependent quantum mechanical wave packet evolution methods. J. Chem. Phys.
**96**, 2077 (1992) ADSCrossRefGoogle Scholar - 18.X. Liu, P. Ding, Dynamic properties of cubic nonlinear Schrödinger equation with varying nonlinear parameter. J. Phys. A, Math. Gen.
**37**, 1589 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar