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Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren, G. R. W. Quispel
TL;DR
The paper addresses the lack of a unified approach for preserving or correctly dissipating energy in PDE discretizations. It presents the average vector-field method as a systematic procedure that achieves these properties across conservative and dissipative PDEs, with formal second-order temporal accuracy.
Problem
A unified way to discuss and enforce energy preservation or correct energy dissipation for PDEs is missing, despite the importance of energy in physical sciences.
Method
The paper applies the average vector-field method after a systematic semi-discretization to conservative and dissipative PDEs with constant structures.
Results
The method preserves semidiscrete energy exactly for conservative PDEs and preserves its monotonic decrease for dissipative PDEs across nonlinear and linear examples.
Takeaways & Limitations
A single geometric integration procedure provides a unified treatment of energy preservation and dissipation for many well-known conservative and dissipative PDEs.
Takeaways & Limitations
The method is formally second order in time, while the relationship between spatial and temporal accuracy is not explored.
Abstract
from arXiv · showhide
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.
1. Introduction
The paper presents a unified procedure for discretizing conservative and dissipative PDEs so that the resulting methods preserve or monotonically dissipate energy. It applies consistent spatial discretization, structure-preserving matrices, and the average vector field method across many linear and nonlinear examples.
- Motivation: The paper frames PDEs as a diverse collection of problems and proposes treating energy conservation and dissipation through one unified computational approach.This contrasts with the relatively compact compendium of methods available for computational ODEs.
- Structure: For Hamiltonian PDEs, a constant skew-adjoint operator preserves the energy because its pairing with the gradient vanishes.The paper denotes the skew-adjoint differential operator by S and the energy by H.
- Structure: For dissipative PDEs, a constant negative semidefinite operator makes H a Lyapunov function whose value decreases monotonically.The operator need not be self-adjoint; the Ginzburg–Landau example uses a non-self-adjoint operator.
- Results: The AVF method preserves the semidiscrete energy exactly in conservative systems and preserves its dissipative structure in dissipative systems.The method extends energy preservation from finite-dimensional vector fields of the form S∇H to the PDE discretizations considered here.
- Method: The proposed procedure discretizes H consistently, replaces the differential structure by a constant skew-symmetric or negative semidefinite matrix, and applies AVF time integration.The resulting semidiscretizations use the standard Euclidean gradient in place of the variational derivative.
- Scope: The framework covers nonlinear and linear conservative and dissipative PDEs, including the examples developed throughout the paper.The paper also notes that AVF is formally second order in time, with higher order available through composition or derivative-based extensions.
2. Conservative PDEs
The paper applies the AVF method to conservative PDEs after spatial semi-discretization, preserving the resulting energy while treating nonlinear and linear examples with a common approach.
- Sine–Gordon equation: The Sine–Gordon equation is semi-discretized using finite differences or spectral differentiation before applying the AVF method.The spectral derivative is computed through a truncated Fourier representation and discrete Fourier transforms.
- Nonlinear conservative PDEs: For Sine–Gordon and Korteweg–de Vries equations, AVF and implicit midpoint have comparable global errors in the reported comparisons.The Sine–Gordon comparisons cover both finite-difference and spectral spatial discretizations; the Korteweg–de Vries integral is evaluated exactly by Simpson’s rule.
- Nonlinear Schrödinger equation: For the nonlinear Schrödinger equation, AVF and midpoint have comparable global errors, but AVF does not conserve total probability because it is not unitary.The AVF integral is evaluated exactly by Simpson’s rule in this example.
- Linear conservative PDEs: The 2D wave equation is discretized with spectral elements and Gauss–Lobatto–Legendre nodes, producing solution snapshots and an energy-error comparison.The reported setup uses six nodes in each spatial direction and compares AVF with ode15s over time.
- Maxwell equations: For Maxwell equations, AVF preserves the discrete energy to machine precision in one dimension and preserves both Hamiltonians and symplectic structures in three dimensions.The three-dimensional result uses two semi-discretizations that produce the same semi-discrete system; in the linear case AVF coincides with midpoint.
3. Dissipative PDEs
The AVF method is applied to dissipative PDEs through a uniform semidiscretization, preserving energy decay or Lyapunov-function decrease across nonlinear and linear examples. In the Ginzburg–Landau example, it avoids the qualitatively incorrect energy increase observed with backward Euler.
- The dissipative examples include the Allen–Cahn, Cahn–Hilliard, Ginzburg–Landau, and heat equations.
- Nonlinear dissipative PDEs: The AVF method preserves the decay of the Ginzburg–Landau energy function, unlike some standard integrators.
- Nonlinear dissipative PDEs: Backward Euler produces a monotonic increase rather than decrease in the Ginzburg–Landau energy, quickly yielding a qualitatively incorrect solution.
- Linear dissipative PDEs: For the heat equation, two discretizations produce identical semidiscrete equations, so the AVF integrator preserves both Lyapunov functions.
- Linear dissipative PDEs: The heat-equation experiments show monotonic decrease of both Lyapunov functions over time.
4. Concluding Remarks
The paper presents the AVF method as a systematic, unified approach for preserving energy in PDE discretizations and preserving its correct dissipation. It also notes broader alternatives for related geometric integration settings.
- The paper provides a systematic and unified AVF-based way to preserve or correctly dissipate energy in PDE discretizations.
- For the treated equations, any energy-preserving B-series method can replace AVF while retaining energy-preservation or dissipation properties.
- Discrete gradient methods can construct geometric integrators for Hamiltonian or non-Hamiltonian PDEs with non-constant matrix D.