Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

Abstract

Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as Korteweg-de Vries, Klein-Gordon, Schrödinger, and Camassa-Holm equations, all with damping/driving terms and time-dependent coefficients. Since key features of the PDEs under consideration are described by local conservation laws, which are independent of the boundary conditions, the proposed (second-order in time) discretizations are developed with the intent of preserving those local conservation laws. The methods are respectively applied to a damped-driven nonlinear Schrödinger equation and a damped Camassa-Holm equation. Numerical experiments illustrate the structure-preserving properties of the methods, as well as favorable results over other competitive schemes.

Multi-Symplectic Method for the Logarithmic-KdV Equation

“…The numerical results are non-dissipative and display excellent conservation properties. In order to overcome the limitation that the multi-symplectic method can only be used in conservative systems, some scholars have further developed this approach and applied it to non-conservation dynamic systems, including the generalized multi-symplectic method [11][12][13], stochastic multi-symplectic method [14,15] and conformal multi-symplectic method [16,17].…”

Nonexistence of local conservation laws for generalized Swift–Hohenberg equation

“…In particular, it follows from Theorem 1 that the original Swift-Hohenberg equation, i.e., (1) with A given by (2), a ≠ 0 , and 2 N∕ u 2 ≠ 0 , has no notrivial local conservation laws. Let us reiterate, however, that Theorem 1 establishes nonexistence of nontrivial local conservation laws for the entire class of generalized Swift-Hohenberg equations of the form (1) with nonconstant polynomial A( ) and nonlinear N(u), which is a far stronger result.…”

“…[2,5,[9][10][11][13][14][15][16] and references therein, even existence of a finite number of conservation laws can be quite helpful in establishing the qualitative behavior of solutions, like e.g. preservation of the solution norm in a certain functional space or of some important physical characteristics like energy or momentum, in the course of time evolution, see for example [1,2,9,11]. Notice that the search for conservation laws is a highly nontrivial task whose complexity grows significantly with the increase of number of independent variables and/or the order of the equation under study, cf.…”

Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

“…The Forward-Backward scheme is the cheapest among the explicit time schemes but also the least accurate.The Runge Kutta integrator RK4 is more accurate and less expensive than RK3. RK-KG (2,6) has the largest computational cost per time iteration but has a good stability. Among the Runge-Kutta integrators considered, it is the least expensive to reach final time.…”

“…The results show that the expected order of accuracy is attained but the computational cost is not considered. Energy conservation properties are studied in [2] on a set of PDEs with damping/driving forces. LERK schemes are shown to be still accurate on non-linear PDEs such as the Korteweg-de Vries equation.…”