Asymptotic stability analysis and error estimate for a class of shallow water wave equation

In this paper, the asymptotic stability and numerical method of shallow water wave equation with Benjamin-Bona-Mahony type is considered. Under suitable assumption, we prove that the solution of the shallow water wave equation is asymptotically convergent to the steady-state solution of the equation, and some exponentially decay rate are obtained. In addition, we also construct a numerical scheme of the equation, we prove that the scheme is unconditionally stable, and we also get the estimate of the full discrete scheme. Finally, some results of the theoretical analysis are verified by numerical experiments


§1 Introduction
Since Benjamin et al. [1] proposed the Benjamini-Bona-Mahony(BBM) model in 1972, the research of this kind of model has been a hot topic.The latest research shows that this kind of model can be used to describe the wave propagation process from deep water to shallow water.The theoretical and numerical research of this kind of model has attracted the interest of researchers.
Medeiros et al. [2] studied the existence and uniqueness of solutions to BBM equation.Amick et al. [3] investigated the long-term behavior of the solution of BBM equation, they used energy estimation, maximum principle, and Cole-Hopf transformation to get the decay rate result of the solution: Biler [4] considered the generalized two-dimensional BBM equation and obtained some estimates of the decay rate: 3 ).
Mei [5] used Fourier transform method and point-by-point method of Green's function to obtain the decay rate estimates for the BBM equation.Chen et al. [6] studied the two-dimensional small amplitude water wave model and obtained some estimation results of the attenuation rate of some solutions.Zhang [7], Guo et al. [8] obtained the decay rate of BBM equation solution in highdimensional space by using low-frequency Fourier method and high-frequency energy method.In the recent research, Kundu et al. [9] obtained the asymptotic attenuation estimation results of BBM equation with homogeneous boundary and the error estimation of some numerical solutions.Numerical aspects: Omrani et al. [10,11] given a second-order numerical scheme for solving BBM equation.The time direction was discretized by Crank-Nicol scheme, and the space direction was discretized by standard finite element method, and the corresponding error estimate was obtained.Dogan [13] used finite element method to solve a class of regular long wave equations, for very small amplitude waves, the algorithm has good accuracy for small amplitude waves.Qin [14] given a numerical scheme for solving BBM equations by using fully discrete mixed finite element method.
In this paper, we will study the asymptotic convergence property and numerical scheme of a class of shallow water wave models, First, we assume that the steady-state equation has a minimum eigenvalue, and by using the energy estimation method, we prove that the solution of BBM shallow-water wave equation converges gradually to the solution of the steady-state equation, and we also get the exponential decay rate estimates under different norms.We also study the numerical format of the equation, i.e. time is discretized by Crank -Nicol Son method, space is discretized by Fourier-Galerkin method, the convergence order of the scheme isO(∆t 2 + N 1−m ), here ∆t is the time discrete step size, N is polynomial order, m is the smoothness of the solution.Finally, we give some numerical examples to verify the correctness of the theoretical analysis.
The structure of this paper is as follows: In the second section, the asymptotic stability analysis of the equation will be given.The third section will discuss the time semi-discrete format.The fourth section will analyze the error estimation of the fully discrete format.Finally, we will give some numerical results.§2 Asymptotic stability analysis We consider the following dissipative BBM shallow water wave equation: satisfy the following initial conditions: and boundary conditions: Here ν is a non-negative constant, ∂ 2 x ∂ t uis the dispersion term, ∂ 2 x uis the dissipative term.§f (x, t) A is a given function.
Here we study the asymptotic properties of the solution of the equation( 1)-(3) when t → ∞.Suppose lim t→∞ u(x, t) = u ∞ , here u ∞ is the steady state solution of the equation, then wheref Consider the weak form of the steady state equation ( 4)-( 5) Now we consider the following assumptions: (A1) Eigenvalue problem Has the smallest positive eigenvalueλ As can be seen from the above equation Using the above inequalities and assumptions A1, we can obtain the existence and uniqueness of solutions to the equation ( 4)- (5).
Based on this, we have the following estimation results of steady-state solutions.Lemma 2.1 Let u ∞ be the solution of ( 4)-( 5), The following estimate holds: Proof: Taking the inner product with u ∞ , we have Then we obtain (9).Using Poincaréinequality, we have That is Here, we will give the asymptotic stability results, that is, we will prove that the difference between the solution of equation ( 1)-( 3) and the solution of steady state equation ( 4)-( 5) is asymptotically convergent with respect to time t.
From ( 8),Young's and Poincaréinequality, we have Using Poincaréinequality again Integrating t on both sides multiplying by e −2δt , noticing a = a 1 + δ, then Theorem 2.3 Under the assumption of A1 and A2, for 0 < a ≤ νλ1 4(λ1+1) , we have x z in equation ( 12) For the first term at the right end, it can be obtained from Young' s inequality: For the second term at the right end, it can be obtained from Young' s inequality and interpolation inequality For the third term at the right end, it can be obtained from Young' s inequality, Poincaré inequality and interpolation inequality: Then From Poincaré inequality, we obtain Integrating t on the both sides and multiplying by e −2δt , that is We consider the following assumptions (A3): Under the assumption that A1 and A3, for a given ε > 0, there is a T > 0 §such that for all t ≥ T , F (t) 0 < ε and the following estimates holds t 0 e as F (s) 2 0 ds.

From lim
t→∞ F (t) 0 = 0, for all ε > 0, there is a T > 0, such that t ≥ T , for F (t) 0 ≤ ε.So we split the right integral term into two parts, (0, T )Ú(T, t), so we have For the (20) equation, integrate on both sides and multiple by e −2at , there are t 0 e as ∂ x z 2 0 ds.
At the same time Then we obtain (21).§3 Stability analysis of numerical schemes In this section, we will give a time semi-discrete scheme and analyze the unconditional stability of this time discrete scheme.Given a positive integerM , let t n = n∆t, n = 0, 1, . . ., M , where ∆t = T /M is the time step size §T be the time.C-N scheme.First step n = 0: When n > 1, we have where Theorem 3.1 The solution of equation ( 22)-( 23) satisfies the following estimation: here Proof: Taking the inner product with 2∆tu 1 in (22).Note that Then Then we obtain the first step of proof.Taking the inner product with 2∆tu n+ 1 2 in (23), we have Then we have We get proof of the theorem.
From taylor expansion and Young' s inequality, we derive