An Implicit Difference Scheme Focusing on the Local Conservation Properties for Burgers Equation

Abstract

Focusing on the local conservation properties, a nine-point implicit difference scheme derived from the multi-symplectic idea, which is named as a generalized multi-symplectic integrator, is presented for the Burgers equation firstly. And then, the associated proofs needed on the local conservation properties of the scheme are given in detail. Finally, to illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the scheme, the numerical experiments on the single-front solution of the Burgers equation by the implicit scheme are reported.

Generalized Multi-Symplectic Method for Dynamic Responses of Continuous Beam Under Moving Load

“…In many cases, the Hamiltonian PDE systems can be written into the form that is very similar to the multi-symplectic form (2.1) except for the skew-symmetry of the coefficient matrices, such as KdV-mKdV-Burgers equation, Burgers equation [Hu et al, 2012a] and the basic vibration equation of the continuous beam under moving load, which can be written into the form…”

“…Moreover, if damping of the structure is considered, the dissipative effect cannot be simulated by these numerical methods. The generalized multi-symplectic method [Hu et al, 2012a;Hu et al, 2013a;Hu et al, 2013b;Hu et al, 2012b] derived from the multi-symplectic idea [Bridges, 1997;Bridges and Reich, 2001;Moore and Reich, 2003a;Moore and Reich, 2003b;Reich, 2000], proved to be a high-efficiency structure-preserving method for analyzing the moving load system and the dissipative problem [Hu et al, 2012a;Hu et al, 2012b], is now used to study the dynamic responses of a nonuniform continuous beam with small damping coefficient under moving loads in this paper. The results of the numerical experiments are presented for the beam under a single load moving at a constant speed or a variable speed.…”

Energy Loss in Pulse Detonation Engine due to Fuel Viscosity

“…Equation 1can be used to describe the flow filed of the transient state as well as the steady state in the detonation tube. The symmetric form of (1) can be obtained referring to the results of [17].…”

“…Actually, the kinematic viscosity coefficient of the fuel ] in the PDE is a positive real number, which makes the symmetric form (2) generalized multisymplectic [17] with external excitation. Following the outline of [17], the local energy error of the symmetric form (2) can be obtained [17]:…”

“…Thus, the energy loss due to the viscosity of the fuel in the PDE is presented by studying the energy loss due to the viscosity of the fluid in Burgers model excited by periodic impulses. Focusing on the energy loss due to the viscosity of the fluid in the PDE, a Burgers model excited by periodic impulses is introduced to describe the flow field in the PDE firstly; and then based on the generalized multisymplectic theory [16][17][18], the expression of the energy loss due to the viscosity of the fluid is derived from the first-order symmetric form of the Burgers model; finally, the numerical results of the energy loss are reported.…”