Numerical preservation of multiple local conservation laws

@article{FrascaCaccia2021NumericalPO,
  title={Numerical preservation of multiple local conservation laws},
  author={Gianluca Frasca-Caccia and Peter E. Hydon},
  journal={Appl. Math. Comput.},
  year={2021},
  volume={403},
  pages={126203}
}

A New Technique for Preserving Conservation Laws

TLDR
A new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws and it is shown that the new technique is practicable for PDEs with three dependent variables.

Optimal parameters for numerical solvers of PDEs

TLDR
A procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step is introduced.

Numerical conservation laws of time fractional diffusion PDEs

This paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy discrete counterparts of these

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

Exponentially fitted methods with a local energy conservation law

A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied

Complete Classification of Local Conservation Laws for Generalized Kuramoto-Sivashinsky Equation

For an arbitrary number of spatial independent variables we present a complete list of cases when the generalized Kuramoto–Sivashinsky equation admits nontrivial local conservation laws of any order,

References

SHOWING 1-10 OF 56 REFERENCES

Simple bespoke preservation of two conservation laws

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All

Locally conservative finite difference schemes for the modified KdV equation

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used

Bespoke finite difference schemes that preserve multiple conservation laws

TLDR
A new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented and is applied to the Korteweg–de Vries equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws.

Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation

By exploiting the fact that conservation laws form the kernel of a discrete Euler operator, we use a recently introduced symbolic-numeric approach to construct a new class of finite difference

Finite difference solutions of the nonlinear Schrödinger equation and their conservation of physical quantities

The solutions of the nonlinear Schrodinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest

Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method

Characteristics of Conservation Laws for Difference Equations

TLDR
The converse of Noether’s Theorem for difference equations is established, the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and all five-point conservation laws for the potential Lotka–Volterra equation are obtained.

A General Framework for Deriving Integral Preserving Numerical Methods for PDEs

A general procedure for constructing conservative numerical integrators for time-dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time

Dissipative or Conservative Finite Difference Schemes for Complex-Valued Nonlinear Partial Different

Abstract We propose a new procedure for designing finite-difference schemes that inherit energy conservation or dissipation property from complex-valued nonlinear partial differential equations
...