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ScienceDirect Procedia Engineering 00 (2017) 000–000 Procedia Engineering 199 (2017) 1539–1543 Procedia Engineering 00 (2017) 000–000

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

X International Conference on Structural Dynamics, EURODYN 2017 X International Conference on Structural Dynamics, EURODYN 2017

Energy Preserving Preserving Finite Finite Difference Difference Scheme Scheme for for Sixth Sixth Order Order Energy Boussinesq Equation Equation Boussinesq a,∗ a,b N. Kolkovska Kolkovskaa,∗ V. Vucheva Vuchevaa,b N. ,, V.

a Institute of Mathematics and Informatics, acad. G. Bonchev st, bl. 8, 1113 Sofia, Bulgaria a Institute of Mathematics and Informatics, acad. G. Bonchev st, bl. 8, 1113 Sofia, Bulgaria b Minning and Geological University ”St. Ivan Rilski”, Student town, 1700 Sofia, Bulgaria b Minning and Geological University ”St. Ivan Rilski”, Student town, 1700 Sofia, Bulgaria

Abstract Abstract We study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conserWe study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conservative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of vative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence in space and time. inc space and time. 2017 The Authors. Published by Elsevier Ltd. c 2017 2017 The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN EURODYN 2017. 2017. Keywords: sixth order Boussinesq equation, finite difference scheme, energy conservation Keywords: sixth47.11.Df, order Boussinesq PACS: 47.35.Fg, 02.60.Lj equation, finite difference scheme, energy conservation PACS: 47.35.Fg, 47.11.Df, 02.60.Lj

1. Introduction 1. Introduction In this paper we consider the Cauchy problem for the nonlinear Boussinesq equation with pure sixth order spatial In this paper we consider the Cauchy problem for the nonlinear Boussinesq equation with pure sixth order spatial derivative (SBE) derivative (SBE) u − u xx − β1 uttxx + β2 u xxxx − β3 u xxxxxx + f (u) xx = 0 − β1 uttxx + β2 u xxxx − β3 u xxxxxx + f (u) xx = 0 utttt − u xx u(0, x) = u (x), u (0, x) = u (x) u(0, x) = u00(x), utt(0, x) = u11(x)

for (t, x) ∈ R+ × R, for (t, x) ∈ R+ × R, for x ∈ R for x ∈ R

(1) (1) (2) (2)

with asymptotic boundary conditions u(x, t) → 0, u xx → 0, u xxxx → 0 for |x| → ∞. Here the dispersion parameters → 0, u → 0 for |x| → ∞. Here the dispersion parameters with asymptotic boundary conditions u(x, t) → 0, u xx 3, .... β1 ≥ 0, β2 0 and β3 > 0 are constants and f (u) = αupp , p = 2,xxxx β1 ≥ 0, β2 0 and β3 > 0 are constants and f (u) = αu , p = 2, 3, .... SBE (1) appears in a number of mathematical models of real processes, for example, in the modelling of longiSBE (1) appears in a number of mathematical models of real processes, for example, in the modelling of longitudinal vibrations in nonlinear atomic chains [5], [8], in some micro-structure problems [1] and in the bidirectional tudinal vibrations in nonlinear atomic chains [5], [8], in some micro-structure problems [1] and in the bidirectional propagation of small amplitude on the surface of shallow water [6]. propagation of small amplitude on the surface of shallow water [6]. For equation (1) with β = 0 two types of solitary waves solutions were numerically found in [5]. These solitary For equation (1) with β11 = 0 two types of solitary waves solutions were numerically found in [5]. These solitary waves are solutions to a nonlinear, fourth order ordinary differential equation (ODE). The exact solution to this ODE waves are solutions to a nonlinear, fourth order ordinary differential equation (ODE). The exact solution to this ODE is found in [1] in terms of Weierstrass elliptic functions. For quadratic or cubic nonlinearity (p = 2 or p = 3) and is found in [1] in terms of Weierstrass elliptic functions. For quadratic or cubic nonlinearity (p = 2 or p = 3) and ∗ ∗

Corresponding author. Tel.: +359-02-979-3868 Corresponding Tel.: +359-02-979-3868 E-mail address:author. [email protected] E-mail address: [email protected]

c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 Peer-reviewunder responsibility of the organizing committee of EURODYN 2017. 1877-7058 © 2017responsibility The Authors. Published by committee Elsevier Ltd. Peer-review under of the organizing of EURODYN 2017. Peer-r

ScienceDirect Procedia Engineering 00 (2017) 000–000 Procedia Engineering 199 (2017) 1539–1543 Procedia Engineering 00 (2017) 000–000

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

X International Conference on Structural Dynamics, EURODYN 2017 X International Conference on Structural Dynamics, EURODYN 2017

Energy Preserving Preserving Finite Finite Difference Difference Scheme Scheme for for Sixth Sixth Order Order Energy Boussinesq Equation Equation Boussinesq a,∗ a,b N. Kolkovska Kolkovskaa,∗ V. Vucheva Vuchevaa,b N. ,, V.

a Institute of Mathematics and Informatics, acad. G. Bonchev st, bl. 8, 1113 Sofia, Bulgaria a Institute of Mathematics and Informatics, acad. G. Bonchev st, bl. 8, 1113 Sofia, Bulgaria b Minning and Geological University ”St. Ivan Rilski”, Student town, 1700 Sofia, Bulgaria b Minning and Geological University ”St. Ivan Rilski”, Student town, 1700 Sofia, Bulgaria

Abstract Abstract We study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conserWe study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conservative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of vative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence in space and time. inc space and time. 2017 The Authors. Published by Elsevier Ltd. c 2017 2017 The TheAuthors. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd. © Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN EURODYN 2017. 2017. Keywords: sixth order Boussinesq equation, finite difference scheme, energy conservation Keywords: sixth47.11.Df, order Boussinesq PACS: 47.35.Fg, 02.60.Lj equation, finite difference scheme, energy conservation PACS: 47.35.Fg, 47.11.Df, 02.60.Lj

1. Introduction 1. Introduction In this paper we consider the Cauchy problem for the nonlinear Boussinesq equation with pure sixth order spatial In this paper we consider the Cauchy problem for the nonlinear Boussinesq equation with pure sixth order spatial derivative (SBE) derivative (SBE) u − u xx − β1 uttxx + β2 u xxxx − β3 u xxxxxx + f (u) xx = 0 − β1 uttxx + β2 u xxxx − β3 u xxxxxx + f (u) xx = 0 utttt − u xx u(0, x) = u (x), u (0, x) = u (x) u(0, x) = u00(x), utt(0, x) = u11(x)

for (t, x) ∈ R+ × R, for (t, x) ∈ R+ × R, for x ∈ R for x ∈ R

(1) (1) (2) (2)

with asymptotic boundary conditions u(x, t) → 0, u xx → 0, u xxxx → 0 for |x| → ∞. Here the dispersion parameters → 0, u → 0 for |x| → ∞. Here the dispersion parameters with asymptotic boundary conditions u(x, t) → 0, u xx 3, .... β1 ≥ 0, β2 0 and β3 > 0 are constants and f (u) = αupp , p = 2,xxxx β1 ≥ 0, β2 0 and β3 > 0 are constants and f (u) = αu , p = 2, 3, .... SBE (1) appears in a number of mathematical models of real processes, for example, in the modelling of longiSBE (1) appears in a number of mathematical models of real processes, for example, in the modelling of longitudinal vibrations in nonlinear atomic chains [5], [8], in some micro-structure problems [1] and in the bidirectional tudinal vibrations in nonlinear atomic chains [5], [8], in some micro-structure problems [1] and in the bidirectional propagation of small amplitude on the surface of shallow water [6]. propagation of small amplitude on the surface of shallow water [6]. For equation (1) with β = 0 two types of solitary waves solutions were numerically found in [5]. These solitary For equation (1) with β11 = 0 two types of solitary waves solutions were numerically found in [5]. These solitary waves are solutions to a nonlinear, fourth order ordinary differential equation (ODE). The exact solution to this ODE waves are solutions to a nonlinear, fourth order ordinary differential equation (ODE). The exact solution to this ODE is found in [1] in terms of Weierstrass elliptic functions. For quadratic or cubic nonlinearity (p = 2 or p = 3) and is found in [1] in terms of Weierstrass elliptic functions. For quadratic or cubic nonlinearity (p = 2 or p = 3) and ∗ ∗

Corresponding author. Tel.: +359-02-979-3868 Corresponding Tel.: +359-02-979-3868 E-mail address:author. [email protected] E-mail address: [email protected]

c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 c 2017 The Authors. Published by Elsevier Ltd. 1877-7058 Peer-reviewunder responsibility of the organizing committee of EURODYN 2017. 1877-7058 © 2017responsibility The Authors. Published by committee Elsevier Ltd. Peer-review under of the organizing of EURODYN 2017. Peer-r