General relaxation methods for initial-value problems with application to multistep schemes
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Numerische Mathematik https://doi.org/10.1007/s00211-020-01158-4
General relaxation methods for initial-value problems with application to multistep schemes Hendrik Ranocha1
· Lajos Lóczi2,3
· David I. Ketcheson1
Received: 6 March 2020 / Revised: 17 October 2020 / Accepted: 20 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples. Mathematics Subject Classification 65L06 · 65L20 · 65M12 · 65M70 · 65P10 · 37M99
1 Introduction Consider an initial-value ordinary differential equation (ODE) in a Banach space: u (t) = f (u(t)) u(0) = u 0 .
B
(1)
Hendrik Ranocha [email protected] Lajos Lóczi [email protected] David I. Ketcheson [email protected]
1
Extreme Computing Research Center (ECRC), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
2
Department of Numerical Analysis, Eötvös Loránd University, Budapest, Hungary
3
Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary
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H. Ranocha et al.
Here and in the following, we use upper indices for t and u to denote the index of the corresponding time step. We say the problem (1) is dissipative with respect to a smooth functional η if d dt
η(u(t)) ≤ 0
(2a)
η (u) f (u) ≤ 0.
(2b)
for all solutions u of (1), i.e. if ∀u :
In the case of equality in (2), we say the problem is conservative. In the numerical solution of dissipative or conservative problems, it is desirable to enforce the same property discretely. For a k-step method we thus require η(u n ) ≤ max{η(u n−1 ), η(u n−2 ), . . . , η(u n−k )}
(3)
for dissipative problems, or η(u n ) = η(u n−1 )
(4)
for conservative problems. A numerical method satisfying this requirement is also said to be dissipative (also known as monotone) or conservative, respectively. For instance, initial-value problems for hyperbolic or parabolic partial differential equations (PDEs) usually have a conserved or dissipated quantity, but in the presence of boundary and/or source terms this quantity may sometimes increase. In that case, energy/entropy estimates are still important and the methods developed in this article are still applicable. 1.1 Related work Conservative or dissipative ODEs arise in a variety of applications and various approaches exist for enforcing these properties discretely; for conservative proble
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