## Spatial discretization of partial differential equations with integrals

Citations: | 2 - 2 self |

### BibTeX

@MISC{Mclachlan_spatialdiscretization,

author = {Robert I. Mclachlan and Robert I. Mclachlan},

title = {Spatial discretization of partial differential equations with integrals},

year = {}

}

### OpenURL

### Abstract

We consider the problem of constructing spatial finite difference approximations on an arbitrary fixed grid which preserve any number of integrals of the partial differential equation and preserve some of its symmetries. A basis for the space of of such finite difference operators is constructed; most cases of interest involve a single such basis element. (The “Arakawa” Jacobian is such an element, as are discretizations satisfying “summation by parts ” identities.) We show how the grid, its symmetries, and the differential operator interact to affect the complexity of the finite difference.

### Citations

974 |
Applications of Lie Groups to Differential Equations
- Olver
- 1993
(Show Context)
Citation Context ... representations of f as needed. (There is another importance difference. If J 2 R n\Thetan satisfies the Jacobi identity and has locally constant rank m, then J automatically has n \Gamma m Casimirs =-=[17]-=-. This need not be true if the Jacobi identity does not hold: there may not be n \Gamma m functions whose gradients span J's nullspace. This is another reason for constructing J's as above that automa... |

418 |
Hyperbolic Systems of Conservation Laws
- Lax
- 1957
(Show Context)
Citation Context ...alysis." (Iserles [3]) Conservative discretizations of partial differential equations have been explored for a long time. What does "conservative" mean? An early definition is due to La=-=x and Wendroff [9]-=-, who considered the class on PDEs with one spatial dimension, u t + @ x (f(u)) = 0 (1.1) and called discretizations of the form u n+1 i \Gamma u n i \Deltat = H(u n i+j ; : : : ; u n i\Gammaj+1 ) \Ga... |

108 |
Computatîonal design for long-term numerical Ïntegratiom of the equations of atmospheric motion
- Arakawa
- 1966
(Show Context)
Citation Context ...uous coefficients [20]. Schemes have also been developed for particular equations that inherit conserved quantities approximating those of the PDE. An early and famous example is the Arakawa Jacobian =-=[1]-=-, a discretization of v x w y \Gamma v y w x which, when applied to the two-dimensional Euler fluid equations, provides two conservation laws corresponding to energy and enstrophy, both quadratic func... |

108 |
A first course in the numerical analysis of differential equations
- Iserles
- 1996
(Show Context)
Citation Context ... affects the complexity of the operators. 1. Conservative discretization "Numerical methods for nonlinear conservation laws are among the great success stories of modern numerical analysis."=-= (Iserles [3]) Conserva-=-tive discretizations of partial differential equations have been explored for a long time. What does "conservative" mean? An early definition is due to Lax and Wendroff [9], who considered t... |

85 |
Generalized hamiltonian dynamics, Phys
- Nambu
- 1973
(Show Context)
Citation Context ...bracket, fF 1 ; : : : ; F p+1 g = K(rF 1 ; : : : ; rF p+1 ) which is multilinear, a derivation in each argument, and completely antisymmetric. Such brackets have been revived in modern times by Nambu =-=[15], who-=-, amongst other things, introduced the 3-bracket on R 3 given by K ijk = " ijk (the alternating tensor). This gives systems of the formsu = K(rI 1 ; rI 2 ) = rI 1 \Theta rI 2 : In particular, the... |

62 |
Finite element and finite difference methods for hyperbolic partial differential equations
- Kreiss, Scherer
- 1974
(Show Context)
Citation Context ...roduct 〈u, v〉 = u t Sv, then D can be used to construct stable finite difference methods, ones that preserve or decrease 〈u, u〉. Such nonEuclidean inner products were introduced by Kreiss and Scherer =-=[12]-=-, and the approach has been recently developed further by Olsson [18] and by Carpenter et al. [5, 2] (see also [3] for a similar approach in the context of spectral methods). 4. Many schemes have been... |

60 | A stable and conservative interface treatment of arbitrary spatial accuracy
- Carpenter, Nordström, et al.
- 1999
(Show Context)
Citation Context ...t preserve or decrease 〈u, u〉. Such nonEuclidean inner products were introduced by Kreiss and Scherer [12], and the approach has been recently developed further by Olsson [18] and by Carpenter et al. =-=[5, 2]-=- (see also [3] for a similar approach in the context of spectral methods). 4. Many schemes have been derived for particular PDEs (e.g. nonlinear wave equations) that preserve nonlinear, even non-quadr... |

46 |
Conservative finite-difference methods on general grids, Symbolic and Numeric Computation Series
- Shashkov
- 1996
(Show Context)
Citation Context ...tegral identities (e.g., Stokes's theorem). In many cases these obey maximum principles and have robust stability properties in difficult situations such as rough grids and discontinuous coefficients =-=[20]-=-. Schemes have also been developed for particular equations that inherit conserved quantities approximating those of the PDE. An early and famous example is the Arakawa Jacobian [1], a discretization ... |

44 |
Summation by parts, projections and stability i
- Olsson
- 1995
(Show Context)
Citation Context ... difference methods, ones that preserve or decrease 〈u, u〉. Such nonEuclidean inner products were introduced by Kreiss and Scherer [12], and the approach has been recently developed further by Olsson =-=[18]-=- and by Carpenter et al. [5, 2] (see also [3] for a similar approach in the context of spectral methods). 4. Many schemes have been derived for particular PDEs (e.g. nonlinear wave equations) that pre... |

43 |
Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups
- Lewis, Simo
- 1994
(Show Context)
Citation Context ...can also have conserved quantities. Even in the ODE example of the free rigid body, the relationship between schemes preserving energy and/or momentum and/or symplectic structure is quite complicated =-=[10]-=-. In this paper we go some way towards uniting these different integrals and different points of view. Our goal is to develop a methodology for building spatial discretizations that preserve discrete ... |

40 |
Differential Forms and Connections
- Darling
- 1994
(Show Context)
Citation Context ... ijk::: @I 1 @x j @I 2 @x k : : : : (2.11) For, suppose K exists. Then I j = f \Delta rI j = 0, so each I j is an integral. Conversely, suppose f has integrals I j . Then (using exterior algebra, see =-=[2]-=-) K = fsrI 1 \Delta \Delta \DeltasrI p det(rI i \Delta rI j ) satisfies (2.11). K is determined uniquely only in the case n = p + 1; see [13] for further details. We write the inner product (2.11) as ... |

37 |
Some remarks concerning Nambu mechanics
- Gautheron
- 1996
(Show Context)
Citation Context ...where the A i are the body's moments of inertia. Contracting against rI 1 gives the standard, Lie-Poisson form of the equations,su i = J ik (u) @I 2 u k = " ijk u j @I 2 @u k : However, later stu=-=dies [5, 23], attempti-=-ng to build a true generalization of Hamiltonian mechanics from such brackets, have found that not all constant K's satisfy the required "fundamental identity" (the analogue of the Jacobi id... |

30 |
Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics
- Simo, Tarnow, et al.
- 1992
(Show Context)
Citation Context ...ly the right generalization of `Hamiltonian' has not yet been found. We are thus reluctantly led to consider only energy-conserving discretizations. Or perhaps we should not be reluctant: Simo et al. =-=[21]-=- have argued and presented detailed evidence from elastodynamics that conserving energy leads to excellent nonlinear stability properties that preserving symplectic structure does not. (Essentially be... |

28 | Geometric integration using discrete gradients
- McLachlan, Quispel, et al.
- 1999
(Show Context)
Citation Context ...ary systemsu = f(u) has integral H. Let z = rH and J ij = f i z j \Gamma f j z i P k z 2 k (2.10) Then J is skew-symmetric, and JrH = f as required. This J is singular at critical points of H, but in =-=[13]-=- it is shown that if f and H are smooth, and the critical points of H are nondegenerate, then there is a smooth matrix J such that f = JrH. This idea extends easily to systems with any number of integ... |

25 | Symplectic integration of Hamiltonian wave equations
- McLachlan
- 1993
(Show Context)
Citation Context ...lly-discrete forms which preserve not only a discrete energy but also a discrete Hamiltonian (symplectic) structure. For systems with canonical Hamiltonian structure, this possibility was explored in =-=[11]-=-. But even before the importance of Hamiltonian PDEs was widely recognized, for which a watershed event was perhaps the 1983 conference [22], it had SPATIAL DISCRETIZATION OF PDES WITH INTEGRALS 3 bee... |

16 |
Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension
- Glassey
- 1992
(Show Context)
Citation Context ...phy, both quadratic functions. It is widely used in computational meteorology. There are many energy-conserving schemes for particular PDEs: Fei and V'asquez [4] for the sine-Gordon equation; Glassey =-=[6]-=- for the Zakharov equations; Glassey and Schaeffer [7] for a nonlinear wave equation. The original presentations of all these are somewhat ad-hoc, the proof of conservation relying on a telescoping su... |

13 |
Solving ODEs numerically while preserving a first integral
- Quispel, Capel
- 1996
(Show Context)
Citation Context ...= (∇H) t A∇H = 0. However, much more is true: given any discretization H of the continuum energy H, an arbitrary system ˙u = f(u) has H as an integral if and only if it can be written in the form (5) =-=[16, 19]-=-. Therefore, the form (5) includes all possible conservative discretizations. This idea extends to systems with any number of integrals: The system of ODEs ˙u = f(u) has integrals I 1 , . . . , I p if... |

12 | Spectral methods on arbitrary grids
- Carpenter, Gottlieb
- 1996
(Show Context)
Citation Context ...rease 〈u, u〉. Such nonEuclidean inner products were introduced by Kreiss and Scherer [12], and the approach has been recently developed further by Olsson [18] and by Carpenter et al. [5, 2] (see also =-=[3]-=- for a similar approach in the context of spectral methods). 4. Many schemes have been derived for particular PDEs (e.g. nonlinear wave equations) that preserve nonlinear, even non-quadratic, energy i... |

9 |
On Foundation of the Generalized Nambu
- Takhtajan
- 1994
(Show Context)
Citation Context ...where the A i are the body's moments of inertia. Contracting against rI 1 gives the standard, Lie-Poisson form of the equations,su i = J ik (u) @I 2 u k = " ijk u j @I 2 @u k : However, later stu=-=dies [5, 23], attempti-=-ng to build a true generalization of Hamiltonian mechanics from such brackets, have found that not all constant K's satisfy the required "fundamental identity" (the analogue of the Jacobi id... |

7 |
Approximately preserving symmetries in the numerical integration of ordinary differential equations
- Iserles, McLachlan, et al.
(Show Context)
Citation Context ...ring the time integration. At the nth time step we use the finite difference tensor sgn(g n )gK, with g n ranging over the symmetries. This decreases the symmetry errors by one power of the time step =-=[8]-=-, which, with \Deltat = (\Deltax) r , may be plenty. Most drastically, K could be only first-order accurate, improving to second through the time integration. This would require a careful stability an... |

7 |
Generating functions for dynamical systems with symmetries
- McLachlan, Quispel
- 1998
(Show Context)
Citation Context ...-Kutta methods such as the midpoint rule; and any number of arbitrary integrals can be preserved by a discrete-time analogue of (3.12) [13]. With one integral, a simple method is based on splitting K =-=[12]-=-. 2. To get simpler finite differences, some of the spatial symmetries can be broken, as for example in the half- and quarter-size Arakawa Jacobians in Fig. 3(b,c). How important is this in practice? ... |

5 |
Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure
- Zeitlin
- 1991
(Show Context)
Citation Context ... Morrison [16] that spatial discretizations of noncanonical Hamiltonian PDEs will not normally be Hamiltonian. One example apart, the curious `sine bracket' Hamiltonian discretization of the Jacobian =-=[24]-=-, this is a difficult and essentially unsolved problem. Probably the right generalization of `Hamiltonian' has not yet been found. We are thus reluctantly led to consider only energy-conserving discre... |

5 | pseudospectral methods, and conservative PDEs
- McLachlan, Robidoux, et al.
- 1999
(Show Context)
Citation Context ...approach also constructs differences on non-constant grids, using the mapping method. We then have to consider the approximation of the continuum conserved quantity H and its derivative. (We refer to =-=[11, 17]-=- for details.) Briefly, let the conserved quantity H = � h(u) du be approximated by a quadrature H = w t h(u), where w is a vector of quadrature weights. Then δH/δu = h ′ (u) ≈ W −1 ∇H, where W = diag... |

5 |
The small dispersion limit for a nonlinear semidiscrete system of equations
- Turner, Rosales
- 1997
(Show Context)
Citation Context ...Burgers’ equation ˙u = 3uux, the semi-discretization ˙u = K ′ ∇H, H = � 1 2 u2 i with K′ given by Eq. (23) is ˙ui = ((ui + ui+1)ui+1 − (ui + ui−1)ui−1) /(2h). This coincides with the well-known (e.g. =-=[21]-=-) result that the semi-discretization conserves � i u2 i ˙ui = 3θ 4h (u2 i+1 − u 2 i−1) + only for θ = 2/3. 3(1 − θ) ui(ui+1 − ui−1) 2h 1 Interestingly, the choice Sij = √ uiuj actually gives a Poisso... |

4 |
High-Order Cyclo-Difference Techniques: An Alternative to Finite Differences
- Carpenter, Otto
(Show Context)
Citation Context ...t preserve or decrease 〈u, u〉. Such nonEuclidean inner products were introduced by Kreiss and Scherer [12], and the approach has been recently developed further by Olsson [18] and by Carpenter et al. =-=[5, 2]-=- (see also [3] for a similar approach in the context of spectral methods). 4. Many schemes have been derived for particular PDEs (e.g. nonlinear wave equations) that preserve nonlinear, even non-quadr... |

3 |
Convergence of a second-order scheme for semilinear hyperbolic equations
- Glassey, Schaeffer
- 1991
(Show Context)
Citation Context ...mputational meteorology. There are many energy-conserving schemes for particular PDEs: Fei and V'asquez [4] for the sine-Gordon equation; Glassey [6] for the Zakharov equations; Glassey and Schaeffer =-=[7]-=- for a nonlinear wave equation. The original presentations of all these are somewhat ad-hoc, the proof of conservation relying on a telescoping sum. The Arakawa Jacobian has the extremely nice propert... |

3 | Skew-adjoint finite difference methods on nonuniform grids
- Kitson, McLachlan, et al.
(Show Context)
Citation Context ...approach also constructs differences on non-constant grids, using the mapping method. We then have to consider the approximation of the continuum conserved quantity H and its derivative. (We refer to =-=[11, 17]-=- for details.) Briefly, let the conserved quantity H = � h(u) du be approximated by a quadrature H = w t h(u), where w is a vector of quadrature weights. Then δH/δu = h ′ (u) ≈ W −1 ∇H, where W = diag... |

2 |
Integral-preserving and skew-adjoint discretizations of partial differential equations
- McLachlan, Robidoux
(Show Context)
Citation Context ...[12] for a discussion. 5. We have deliberately avoid mentioning boundaries and the precise degree of smoothness required of the arguments that make K skew. 22 ROBERT I. MCLACHLAN These are studied in =-=[14]-=-. If the PDE develops shocks, a careful weighting of the F (i) will be required to capture them well, the analogue of the many methods for choosing H in (1.2) [3]. The present work applies to infinite... |

1 |
Two energy-conserving numerical schemes for the SineGordon equation
- Fei, Vasquez
- 1991
(Show Context)
Citation Context ...on laws corresponding to energy and enstrophy, both quadratic functions. It is widely used in computational meteorology. There are many energy-conserving schemes for particular PDEs: Fei and V'asquez =-=[4]-=- for the sine-Gordon equation; Glassey [6] for the Zakharov equations; Glassey and Schaeffer [7] for a nonlinear wave equation. The original presentations of all these are somewhat ad-hoc, the proof o... |

1 |
Princeton Plasma Physics Laboratory Report PPL–1783
- Morrison
- 1981
(Show Context)
Citation Context ...rtance of Hamiltonian PDEs was widely recognized, for which a watershed event was perhaps the 1983 conference [22], it had SPATIAL DISCRETIZATION OF PDES WITH INTEGRALS 3 been pointed out by Morrison =-=[16]-=- that spatial discretizations of noncanonical Hamiltonian PDEs will not normally be Hamiltonian. One example apart, the curious `sine bracket' Hamiltonian discretization of the Jacobian [24], this is ... |