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Networks and Heterogeneous Media

2014, Volume 9, Issue 3: 383-416. doi: 10.3934/nhm.2014.9.383
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Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree

  • 1.
    University of Maryland, Department of Mathematics, College Park, MD 20742-4015
  • Received: 01 February 2014 Revised: 01 June 2014

  • Primary: 65C50, 60J27; Secondary: 60J28.

  • The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap has become a popular theme. We consider stochastic networks representing potential energy landscapes whose states and edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. For the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range. However, if the observation time is limited, the expected spectral gap appears.

    Citation: Maria Cameron. Computing the asymptotic spectrum for networks representing energy landscapesusing the minimum spanning tree[J]. Networks and Heterogeneous Media, 2014, 9(3): 383-416. doi: 10.3934/nhm.2014.9.383

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  • The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap has become a popular theme. We consider stochastic networks representing potential energy landscapes whose states and edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. For the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range. However, if the observation time is limited, the expected spectral gap appears.


    1.   Introduction

    Nonlinear partial differential equations (PDEs) are often used to describe some significant problems in natural science and engineering technology. The analysis of nonlinear waves has garnered increasing attention in the fields of shallow water, plasma, nonlinear optics, Bose−Einstein condensates, and fluids. The nonlinear PDEs in these fields can be solved using lump solutions. The lump solution and interaction hybrid solutions were first discovered by Zakharov in [1] and Craik in [2], respectively. These special solutions are of great significance for the study of nonlinear integrable equations [3,4,5]. To obtain these analytical solutions, the scholars have proposed a variety of methods, including Ansatz technique [6], Hirota's bilinear method (HBM) [7], and the inverse scattering transform (IST) [8]. The HBM proposed by Hirota, has been widely used as an effective approach to study the nonlinear dynamics wave equation, resulting in numerous richer solutions, such as solitons solutions, novel breather waves, lump solutions, two-wave solutions, and rogue wave solutions ([9,10,11,12,13,14]). These solutions are visualized in three-dimensional graphics via numerical simulations, making it easier to understand the propagation of nonlinear waves.

    The study of the stability of integrable equations, such as the KPI equation, the Ishimori equation, the nonlinear Schrödinger (NLS) equation and the KdV equation have further enriched the theory of nonlinear wave equations. Spectral methods can effectively solve the problem of linear stability. In [15], Degasperis et al. proposed the construction of the eigenmodes of the linearized equation using the associated Lax pair and provided the computation of both analytical and numerical solutions with the example of two coupled NLS equations. In [16], Ablowitz provided a comprehensive review of research methods for integrability and nonlinear waves, including Bäcklund transformations, Darboux transformations, direct integral equations or Riemann−Hilbert or Dbar methods, and HBM. By combining the KdV and NLS equations, the author further elaborated on the ideas and background of the IST method.

    The computation of nonlinear PDEs has become a very active research topic. With advancements in traditional methods like finite element ([17,18]), finite difference ([19,20,21,22,23,24]), finite volume ([25,26,27]), and spectral methods ([28,29,30]), numerous outstanding research results have been achieved in the numerical approximations of nonlinear PDEs. In [31], the authors developed a time-two-grid difference scheme for nonlinear Burgers equations. In [32], a method combining the barycentre Lagrange interpolation collocation technique with a second-order operator splitting approach was proposed for the purpose of solving the NLS equation. Based on novel shifted Delannoy functions, Ansari et al. [33] employed a matrix collocation technique to numerically approximate the singularly perturbed parabolic convection−diffusion−reaction problems.

    In recent years, for the treatment of nonlinear terms, there have been a lot of unconditionally energy dissipative numerical schemes for Allen−Cahn and Cahn−Hilliard gradient flows models, such as:

    (ⅰ) CSS (convex splitting) scheme [34,35,36];

    (ⅱ) IMEX (stabilized semi-implicit) scheme [37,38];

    (ⅲ) ETD (exponential time differencing) scheme [39,40];

    (ⅳ) IEQ (invariant energy quadratization) scheme [41];

    (ⅴ) SAV (scalar auxiliary variable) scheme [42,43];

    (ⅵ) E-SAV (exponential scalar auxiliary variable) scheme [44].

    The idea of (ⅰ) is to decompose the energy function into convex and concave parts, handling the convex part implicitly and the concave part explicitly. The advantage is that it can achieve second-order unconditional stability. However, the drawback is that it still requires solving nonlinear equations. The method of (ⅱ) yields extra error, which makes it difficult to construct a higher-order scheme. The (ⅳ) and (ⅴ) approaches make it easier to handle nonlinear terms by defining auxiliary variables that transform the nonlinear potential function into a simple quadratic form. Nevertheless, an inner product must be calculated before obtaining the next time value. Compared with (ⅳ) and (ⅴ), the variable defined in (ⅵ) does not require any assumptions. And the E-SAV method can easily construct an explicit scheme that can preserve energy stability. But this physical property can not be satisfied with the explicit SAV scheme.

    In addition, Huang et al. [45] studied a new SAV method to approximate the gradient flows, which is an improvement of the SAV method. By defining an auxiliary variable in the new SAV method as a shifted total energy function, instead of focusing on the nonlinear parts of the classic SAV method, we replace the dynamic equation for that variable with the energy balance equation of the gradient flow. This facilitates the construction of high-order and energy-stable discrete schemes. In [46], Liu et al. further developed an exponential semi-implicit scalar auxiliary variable (ESI-SAV) method for the phase field equation. The ESI-SAV method can preserve the advantages of both the new SAV and E-SAV methods and be applied more effectively to other dissipative systems.

    On the basis of the above methods, researchers began to apply the above nonlinear processing techniques to approximate semi-linear wave equations. Jiang et al. [47] proposed an IEQ approach and established an energy-preserving linear implicit scheme for the sine-Gordon equation. In [48], based on the SAV method with a combination of the Gauss technique and the extrapolation method, Li et al. provided a high-order energy-conserving and linearly implicit scheme. Wang et al. [49] developed a second-order SAV Fourier spectral method to solve a nonlinear fractional generalized wave equation.

    The primary objective of this paper is to develop second-order and Hamiltonian conserved semi-discrete schemes for semi-linear wave equations. Following the superiorities of the E-SAV and ESI-SAV methods, two different numerical schemes are given by utilizing the Crank−Nicolson scheme for temporal approximation. Furthermore, the convergence order and the evolution curve of the Hamiltonian function are validated through numerical experiments.

    The paper is organized as follows: In Section 2, by introducing scalar auxiliary variables, we obtain two equivalent forms for the semi-linear wave equation in the continuous case. In Sections 3 and 4, by using the Crank−Nicolson scheme, we propose two semi-discrete schemes corresponding to the equivalent forms and provide proof of the convergence order. Two numerical examples are implemented to test the effectiveness of the theoretical analysis in Section 5.

    2.   Preliminary results

    We consider the following semi-linear wave equation:

    $ {ytt−Δy−f(y)=0,x∈Ω,t∈(0,T),y(x,t)=0,x∈∂Ω,t∈(0,T),y(x,0)=y0(x),yt(x,0)=g(x),x∈Ω, $ (2.1)

    where $ \Omega \subset \mathbb{R}^{d} (d = 2, 3) $ is a bounded convex domain, $ T > 0 $ is a fixed number, $ f(y) = -F'(y) $ is a nonlinear term, and $ F(y) $ satisfies $ F(y)\in\mathcal{C}^{3}(R) $. The Hamiltonian function is defined as

    $ H(y)=12||yt||2+12||∇y||2+∫ΩF(y)dx. $ (2.2)

    Proposition 2.1. The system (2.1) holds the following Hamiltonian conservation law:

    $ dH(y)dt=0. $ (2.3)

    Proof. Multiplying the first equation of the system (2.1) by $ y_{t} $ yields

    $ yttyt−Δyyt−f(y)yt=0. $ (2.4)

    By using the continuous Leibniz rule, we obtain

    $ yttyt=12((yt)2)t,−f(y)yt=(F(y))t, $ (2.5)

    and

    $ Δyyt=(yx1x1+yx2x2)yt=(yx1yt)x1+(yx2yt)x2−yx1(yt)x1−yx2(yt)x2=(yx1yt)x1+(yx2yt)x2−12(y2x1+y2x2)t. $ (2.6)

    Inserting (2.5) and (2.6) into (2.4), we obtain

    $ [12((yt)2)t+12(y2x1+y2x2)t+(F(y))t]−[(yx1yt)x1+(yx2yt)x2]=0. $ (2.7)

    Integrating (2.7) over the spatial domain $ \Omega $ and combining the boundary conditions leads to

    $ ∫Ω[12((yt)2)t+12(y2x1+y2x2)t+(F(y))t]dx=0, $ (2.8)

    From (2.8) and (2.2), we can deduce that

    $ dH(y)dt=∫Ω[12((yt)2)t+12(y2x1+y2x2)t+(F(y))t]dx=0. $ (2.9)

    The proof is complete. □

    We introduce two scalar auxiliary variables as

    $ \omega = y_{t},\; r(t) = \int_{\Omega}F(y)d\boldsymbol{x} = \mathcal{H}_{2}(y). $

    Then the system (2.1) can be equivalently rewritten as

    $ ω=yt,x∈Ω,t∈(0,T), $ (2.10)
    $ ωt−Δy−qy,rf(y)=0,x∈Ω,t∈(0,T), $ (2.11)
    $ rt=−qy,r(f(y),ω),x∈Ω,t∈(0,T), $ (2.12)
    $ y(x,t)=0,ω(x,t)=0,x∈∂Ω,t∈(0,T), $ (2.13)
    $ y(x,0)=y0(x),ω(x,0)=g(x),r(0)=H2(y0(x)),x∈Ω, $ (2.14)

    where

    $ qy,r=exp{r}exp{H2(y)}. $

    For the system (2.10)–(2.14), we have the following lemma:

    Lemma 2.1. (Hamiltonian conservation) The above system (2.10)–(2.14) satisfies

    $ d˜H(y)dt=0, $ (2.15)

    where the modified Hamiltonian function

    $ ˜H(t)=12||ω||2+12||∇y||2+r. $ (2.16)

    Proof. Taking the inner product of (2.11) by $ \omega $, then combining (2.10) and (2.12), we obtain

    $ (ωt,ω)+(∇y,∇yt)+rt=0. $

    Obviously, we can deduce

    $ ddt(12||ω||2+12||∇y||2+r)=0. $

    The proof is complete. □

    Next, introducing a new variable

    $ \mathcal{R}(t) = H(y), $

    which satisfies the dissipation law

    $ dRdt=dH(y)dt=0, $ (2.17)

    and defining $ \xi = \frac{\exp (\mathcal{R})}{\exp (H(y))} $, we can know that $ \xi\equiv1 $ in the continuous case. Replacing the factor $ q^{y, r} $ in (2.11) by $ Q(\xi) $, where $ Q(\xi) $ is a polynomial function of $ \xi $, the system (2.1) can be transformed into the following equivalent form:

    $ ω=yt,x∈Ω,t∈(0,T), $ (2.18)
    $ ωt−Δy−Q(ξ)f(y)=0,x∈Ω,t∈(0,T), $ (2.19)
    $ ξ=exp(R)exp(H(y))x∈Ω,t∈(0,T), $ (2.20)
    $ Q(ξ)=ξ(2−ξ)x∈Ω,t∈(0,T), $ (2.21)
    $ dRdt=0,x∈Ω,t∈(0,T), $ (2.22)
    $ y(x,t)=0,ω(x,t)=0,x∈∂Ω,t∈(0,T), $ (2.23)
    $ y(x,0)=y0(x),ω(x,0)=g(x),R(0)=H(y0),x∈Ω. $ (2.24)

    Furthermore, (2.17) and (2.22) imply that the system (2.18)–(2.24) is Hamiltonian conserved.

    3.   CN-E-SAV scheme

    Let $ 0 = t_{0} < t_{1} < ... < t_{N} = T $ be a uniform partition of the time interval $ [0, T] $ with the time steps $ \Delta t = T/N $ and $ t^{n} = n\Delta t $. Then, utilizing the Crank−Nicolson scheme to discretize the system (2.10)–(2.14), a second-order Carnk−Nicolson E-SAV (CN-E-SAV) scheme can be formulated as follows:

    $ ωn+12=1Δt(yn+1−yn), $ (3.1)
    $ 1Δt(ωn+1−ωn)−Δyn+12−ˆqn+12f(ˆyn+12)=0, $ (3.2)
    $ 1Δt(rn+1−rn)=−(ˆqn+12f(ˆyn+12),ωn+12), $ (3.3)

    where

    $ ωn+12=12(ωn+1+ωn),yn+12=12(yn+1+yn), $

    and $ \hat{q}^{n+\frac{1}{2}} > 0 $ with $ (\hat{y}^{n+\frac{1}{2}}, \hat{r}^{n+\frac{1}{2}}) $ being generated by the first-order scheme with the time step size $ \dfrac{\Delta t}{2} $, i.e.,

    $ ˆωn+12=1Δt/2(ˆyn+12−yn), $ (3.4)
    $ 1Δt/2(ˆωn+12−ωn)−Δˆyn+12−qnf(yn)=0, $ (3.5)
    $ 1Δt/2(ˆrn+12−rn)=−qn(f(yn),ˆωn+12). $ (3.6)

    In order to better understand this scheme, by plugging (3.1) into (3.2) and (3.4) into (3.5), respectively, we can get

    $ 2Δt2yn+1−12Δyn+1=2Δt2yn+2Δtωn+12Δyn+ˆqn+12f(ˆyn+12), $ (3.7)
    $ 4Δt2ˆyn+12−Δˆyn+12=4Δt2yn+2Δtωn+qnf(yn). $ (3.8)

    So we can implement the CN-E-SAV scheme as follows:

    $ (i). Computeqnfromqn=exp{rn}exp{H2(yn)};(ii). Computeˆyn+12from(3.8);(iii). Computeˆωn+12from(3.4);(iv). Computeˆrn+12from(3.6);(v). Computeˆqn+12fromˆqn+12=exp{ˆrn+12}exp{H2(ˆyn+12)};(vi). Computeyn+1from(3.7);(vii). Computeωn+1from(3.1);(viii). Computern+1from(3.3). $

    3.1.   Hamiltonian conservation

    We show the conservation of the CN-E-SAV scheme by the following theorem:

    Theorem 3.1. The CN-E-SAV scheme (3.1)–(3.3) is Hamiltonian conserved:

    $ Hn+1=HnwithHn=12||ωn||2+12||∇yn||2+rn. $ (3.9)

    Proof. Taking the inner product with (3.1) and (3.2) by $ y^{n+\frac{1}{2}} $, $ \omega^{n+\frac{1}{2}} $, we have

    $ (ωn+12,yn+12)=12Δt||yn+1||2−12Δt||yn||2, $ (3.10)

    and

    $ 12||ωn+1||2−12||ωn||2+Δt(∇yn+12,∇ωn+12)−ˆqn+12(f(ˆyn+12),ωn+12)Δt=0. $ (3.11)

    Substituting (3.10) into (3.11) and combining (3.3), we obtain

    $ (12||ωn+1||2+12||∇yn+1||2+rn+1)−(12||ωn||2+12||∇yn||2+rn)=0. $

    The proof is complete. □

    3.2.   The error of the CN-E-SAV scheme

    We will follow the next two steps to complete the error estimation of the CN-E-SAV scheme:

    step (ⅰ): complete the error between (2.10)–(2.14) and (3.4)–(3.6);

    step (ⅱ): with the help of the results of step (ⅰ), the error of (2.10)–(2.14) and (3.1)–(3.3) is further estimated.

    For simplicity, we define

    $ ˆen+12ω=ω(tn+12)−ˆωn+12,enω=ω(tn)−ωn,ˆen+12y=y(tn+12)−ˆyn+12,eny=y(tn)−yn,ˆen+12r=r(tn+12)−ˆrn+12,enr=r(tn)−rn. $

    It follows from (2.10)–(2.14) that the exact solution $ (\omega, y, r) $ satisfies

    $ ω(tn+12)=1Δt/2(y(tn+12)−y(tn))−1Δt/2ˆTny1, $ (3.12)
    $ ω(tn+12)−ω(tn)Δt/2−Δy(tn+12)−q(tn)f(y(tn))=1Δt/2ˆTnω−Δ(y(tn+12)−y(tn)), $ (3.13)
    $ r(tn+12)−r(tn)=−(q(tn)f(y(tn)),y(tn+12)−y(tn)−ˆTny2)+ˆTnr, $ (3.14)

    where the truncation functions are defined as

    $ ˆTnω=∫tn+12tn(tn+12−t)ωttdt,ˆTny1=∫tn+12tn(tn−t)yttdt,ˆTny2=∫tn+12tn(tn+12−t)yttdt,ˆTnr=∫tn+12tn(tn+12−t)rttdt, $

    and satisfy

    $ ||ˆTnω||2≤C(Δt)3∫tn+12tn||ωtt||2dt,||ˆTny1||2≤C(Δt)3∫tn+12tn||ytt||2dt, $ (3.15)
    $ ||ˆTny2||2≤C(Δt)3∫tn+12tn||ytt||2dt,|ˆTnr|2≤C(Δt)3∫tn+12tn|rtt|2dt. $ (3.16)

    For the purposes of theorem proving, we present the following assumptions and lemmas:

    Assumption 3.1. There exist constants $ Q^{*} $, $ Q_{*} $, $ \tilde{Q}^{*} $, and $ \tilde{Q}_{*} $ independent of $ \Delta t $ such that $ Q_{*} \leq q^{n}\leq Q^{*} $, $ \tilde{Q}_{*} \leq\hat{q}^{n+\frac{1}{2}}\leq \tilde{Q}^{*} $ for all n.

    Lemma 3.1. Denote

    $ A=q(tn)f(y(tn))−qnf(yn),B=ˆqn+12f(ˆyn+12)−q(tn+12)f(y(tn+12)), $

    then the following holds:

    $ ||A||≤C(||∇eny||+|enr|),||B||≤C(||∇ˆen+12y||+|ˆen+12r|), $

    where the constant $ C > 0 $ depends on $ Q^{*}, Q_{*}, \tilde{Q}^{*} $, $ \tilde{Q}_{*}, |\Omega|, y_{0} $, and $ ||f||_{C^{1}(R)} $.

    Proof. Similar to the proof of Lemma 4 of [51], the results can be proved by applying Poincaré's inequality to $ ||e_{y}^{n}||, ||\hat{e}_{y}^{n+\frac{1}{2}}|| $ once more. □

    Lemma 3.2. Assume $ \Delta t $ is sufficiently small and satisfies

    $ yt∈L∞(0,T;L2)∩L2(0,T;H2),ytt∈L2(0,T;L2)∩L2(0,T;H1),yttt∈L2(0,T;L2). $

    Then, we have

    $ ||ˆen+12ω||2+||∇ˆen+12y||2+|ˆen+12r|2≤ˇC(Δt)3∫tn+12tn(||ωtt||2+||∇ytt||2+|rtt|2+||ytt||2+||yt||2H2(Ω))ds+ˇC(||enω||2+||∇eny||2+|enr|2). $ (3.17)

    Proof. Subtracting (3.4)–(3.6) from (3.12)–(3.14), we have the error equations as

    $ ˆen+12ω=1Δt/2(ˆen+12y−eny)−1Δt/2ˆTny1, $ (3.18)
    $ ˆen+12ω−enωΔt/2−Δˆen+12y=q(tn)f(y(tn))−qnf(yn)−Δ(y(tn+12)−y(tn))+1Δt/2ˆTnω, $ (3.19)
    $ ˆen+12r−enr=−(qnf(yn),ˆen+12y−eny)+(qnf(yn)−q(tn)f(y(tn)),y(tn+12)−y(tn))+(q(tn)f(y(tn)),ˆTny2)+ˆTnr. $ (3.20)

    Taking the inner product of (3.19) with $ \hat{e}_{\omega}^{n+\frac{1}{2}} $ and combining (3.18), we have

    $ ||ˆen+12ω||2−||enω||2+||ˆen+12ω−enω||2+||∇ˆen+12y||2−||∇eny||2+||∇(ˆen+12y−eny)||2=Δt(A,ˆen+12ω)−Δt(Δ(y(tn+12)−y(tn)),ˆen+12ω)+2(ˆTnω,ˆen+12ω)−2(ˆTny1,Δˆen+12y):=Λ1+Λ2+Λ3+Λ4. $ (3.21)

    For the first term on the right-hand side of (3.21), according to Lemma 3.1, we obtain

    $ Λ1≤C(Δt)2||A||2+14||ˆen+12ω||2≤˜C1(Δt)2(||∇eny||2+|enr|2)+14||ˆen+12ω||2. $ (3.22)

    Then for $ \Lambda_{2}, \; \Lambda_{3}, \; \Lambda_{4} $, using Young's inequality and (3.15), one has

    $ Λ2≤C(Δt)3∫tn+12tn||yt||2H2(Ω)ds+14||ˆen+12ω||2, $ (3.23)
    $ Λ3≤C(Δt)3∫tn+12tn||ωtt||2dt+14||ˆen+12ω||2, $ (3.24)
    $ Λ4≤C(Δt)3∫tn+12tn||∇ytt||2dt+14||∇ˆen+12y||2. $ (3.25)

    Combining (3.22)–(3.25) with (3.21), we can have

    $ ||ˆen+12ω||2+4||ˆen+12ω−enω||2+||∇ˆen+12y||2+4||∇(ˆen+12y−eny)||2≤4˜C1(Δt)2(||∇eny||2+|enr|2)+4(||enω||2+||∇eny||2)+C(Δt)3∫tn+12tn(||ωtt||2+||∇ytt||2+||yt||2H2(Ω))ds. $ (3.26)

    Multiplying (3.20) by $ \hat{e}_{r}^{n+\frac{1}{2}} $, we can obtain

    $ |ˆen+12r|2−|enr|2+|ˆen+12r−enr|2=−2(qnf(yn),ˆen+12y−eny)ˆen+12r−2(A,y(tn+12)−y(tn))ˆen+12r+2(q(tn)f(y(tn)),ˆTny2)ˆen+12r+2ˆTnrˆen+12r:=Ψ1+Ψ2+Ψ3+Ψ4. $ (3.27)

    According to $ q^{n}\leq Q^{*}, \; F\in \mathcal{C}^{3}(R) $, and Poincaré's inequality, we can derive the following estimate for $ \Psi_{1} $:

    $ Ψ1≤||qnf(yn)||||ˆen+12y−eny|||ˆen+12r|≤C∗||∇(ˆen+12y−eny)||2+14|ˆen+12r|2. $ (3.28)

    The estimate for $ \Psi_{2} $ is similar to (3.22), so we obtain

    $ Ψ2≤Δt||A||||y(tn+12)−y(tn)Δt|||ˆen+12r|≤˜C2(Δt)2(||∇eny||2+|enr|2)+14|ˆen+12r|2. $ (3.29)

    For the last two terms of (3.27), by utilizing (3.16), we have that

    $ Ψ3+Ψ4≤C(Δt)3∫tn+12tn(|rtt|2+||ytt||2)ds+14|ˆen+12r|2. $ (3.30)

    Therefore, injecting (3.28)–(3.30) into (3.27) leads to

    $ |ˆen+12r|2+4|ˆen+12r−enr|2≤4˜C2(Δt)2(||∇eny||2+|enr|2)+C(Δt)3∫tn+12tn(|rtt|2+||ytt||2)dt+4C∗||∇(ˆen+12y−eny)||2+4|enr|2. $ (3.31)

    Multiplying (3.26) by $ C^{*} $ and adding it to (3.31) implies

    $ C∗||ˆen+12ω||2+C∗||∇ˆen+12y||2+|ˆen+12r|2≤4(˜C1C∗+˜C2)(Δt)2(||∇eny||2+|enr|2)+4C∗(||enω||2+||∇eny||2)+4|enr|2+C(Δt)3∫tn+12tn(||ωtt||2+||∇ytt||2+|rtt|2+||ytt||2+||yt||2H2(Ω))ds. $

    The proof is completed when $ C^{*}\geq1 $ and $ (\Delta t)^{2}\leq\frac{C^{*}}{\tilde{C}_{1}C^{*}+\tilde{C}_{2}} $. □

    Next, we will derive the convergence order of the CN-E-SAV scheme (3.1)–(3.3). Clearly, the exact solution $ (\omega, y, r) $ satisfies

    $ ω(tn+1)+ω(tn)2=1Δt(y(tn+1)−y(tn))+Rnω−1ΔtTny, $ (3.32)
    $ 1Δt(ω(tn+1)−ω(tn))−Δy(tn+1)+y(tn)2−q(tn+12)f(y(tn+12))=1ΔtTnω−ΔRny, $ (3.33)
    $ 1Δt(r(tn+1)−r(tn))=−(q(tn+12)f(y(tn+12)),ω(tn+1)+ω(tn)2−Rω)+1ΔtTnr, $ (3.34)

    where the truncation functions are defined by

    $ Tω=ω(tn+1)−ω(tn)−Δtωt(tn+12),Ty=y(tn+1)−y(tn)−Δtyt(tn+12),Tr=r(tn+1)−r(tn)−Δtrt(tn+12),Rω=ω(tn+1)+ω(tn)2−ω(tn+12),Ry=y(tn+1)+y(tn)2−y(tn+12),Rr=r(tn+1)+r(tn)2−r(tn+12). $

    The truncation functions satisfy the following lemma:

    Lemma 3.3. ([50]) The following estimates hold

    $ ||Tω||2≤C(Δt)5∫tn+1tn||ωttt||2ds,||Ty||2≤C(Δt)5∫tn+1tn||yttt||2ds,|Tr|2≤C(Δt)5∫tn+1tn|rttt|2ds,||Rω||2≤C(Δt)3∫tn+1tn||ωtt||2ds,||Ry||2≤C(Δt)3∫tn+1tn||ytt||2ds,|Rr|2≤C(Δt)3∫tn+1tn|rtt|2ds. $

    Theorem 3.2. Let $ (\omega(t_{n}), y(t_{n}), r(t_{n})) $ and $ (\omega^{n}, y^{n}, r^{n}) $ be the solutions of (2.10)–(2.14) and CN-E-SAV scheme (3.1)–(3.3), respectively. Suppose that the assumptions in Lemma 3.2 hold and assume further

    $ ytt∈L2(0,T;L2)∩L2(0,T;H2),yttt∈L2(0,T;L2)∩L2(0,T;H1),ytttt∈L2(0,T;L2). $

    Then, we have

    $ ||enω||2+||∇eny||2+|enr|2≤C(Δt)4. $

    Proof. Subtracting (3.1)–(3.3) from (3.32)–(3.34), we derive the following error equations:

    $ en+12ω=1Δt(en+1y−eny)+Rnω−1ΔtTny, $ (3.35)
    $ 1Δt(en+1ω−enω)−Δen+12y=q(tn+12)f(y(tn+12))−ˆqn+12f(ˆyn+12)+1ΔtTnω−ΔRny, $ (3.36)
    $ 1Δt(en+1r−enr)=(ˆqn+12f(ˆyn+12)−q(tn+12)f(y(tn+12)),ω(tn+1)+ω(tn)2)−(ˆqn+12f(ˆyn+12),en+12ω)+(q(tn+12)f(y(tn+12)),Rnω)+1ΔtTnr. $ (3.37)

    Taking the inner product of (3.36) with $ e_{\omega}^{n+\frac{1}{2}} $ and combining (3.35), we have

    $ 12||en+1ω||2−12||enω||2+12||∇en+1y||2−12||∇eny||2=−Δt(B,en+12ω)+(Tnω,en+12ω)−Δt(ΔRny,en+12ω)−Δt(∇en+12y,∇Rnω)−Δt(∇en+12y,∇Tny):=Υ1+Υ2+Υ3+Υ4+Υ5. $ (3.38)

    According to Lemma 3.1, we obtain

    $ Υ1≤CΔt(|ˆen+12r|2+||∇ˆen+12y||2+||en+12ω||2). $ (3.39)

    For the last four terms of (3.38), it follows from Lemma 3.3 that

    $ Υ2≤C(Δt)4∫tn+1tn||ωttt||2dt+CΔt||en+12ω||2, $ (3.40)
    $ Υ3≤C(Δt)4∫tn+1tn||Δytt||2dt+CΔt||en+12ω||2, $ (3.41)
    $ Υ4≤C(Δt)4∫tn+1tn||∇ωtt||2dt+CΔt||∇en+12y||2, $ (3.42)
    $ Υ5≤C(Δt)4∫tn+1tn||∇yttt||2dt+CΔt||∇en+12y||2. $ (3.43)

    Substituting (3.39)–(3.43) into (3.38), we can obtain

    $ 12||en+1ω||2−12||enω||2+12||∇en+1y||2−12||∇eny||2≤CΔt(|ˆen+12r|2+||∇ˆen+12y||2)+CΔt(||en+12ω||2+||∇en+12y||2)+C(Δt)4∫tn+1tn(||ωttt||2+||∇yttt||2+||Δytt||2+||∇ωtt||2)dt. $ (3.44)

    Multiplying (3.3) by $ e_{r}^{n+\frac{1}{2}} $, we obtain

    $ 12(|en+1r|2−|enr|2)=Δten+12r(B,ω(tn+1)+ω(tn)2)−Δten+12r(ˆqn+12f(ˆyn+12),en+12ω)+Δt(q(tn+12)f(y(tn+12)),Rnω)en+12r+Tnren+12r:=Φ1+Φ2+Φ3+Φ4. $ (3.45)

    Next, we estimate the right-hand side of (3.45). Applying Lemma 3.1, Lemma 3.3, and Young's inequality, we deduce that

    $ Φ1≤CΔt|en+12r|||B||||ω(tn+1)+ω(tn)2||≤CΔt(|ˆen+12r|2+||∇ˆen+12y||2+|en+12r|2), $ (3.46)
    $ Φ2≤CΔt(|en+12r|2+||en+12ω||2), $ (3.47)
    $ Φ3+Φ4≤C(Δt)4∫tn+1tn(||ωtt||2+|rttt|2)dt+CΔt|en+12r|2. $ (3.48)

    Combining the above estimates of (3.46)–(3.48) together, one has

    $ 12|en+1r|2−12|enr|2≤CΔt(|ˆen+12r|2+||∇ˆen+12y||2)+CΔt(||en+12ω||2+|en+12r|2)+C(Δt)4∫tn+1tn(||ωtt||2+|rttt|2)dt. $ (3.49)

    By adding (3.44) and (3.49) and summing the index $ k $ from $ 0 $ to $ n-1 $, we obtain

    $ ||enω||2+||∇eny||2+||enr||2≤CΔtn−1∑k=0(||ekω||2+||∇eky||2+|ekr|2)+CΔtn−1∑k=0(|ˆek+12r|2+||∇ˆek+12y||2)+C(Δt)4∫tn0(||ωttt||2+|rttt|2+||∇yttt||2+||ωtt||2+||ytt||2+|rtt|2+||Δytt||2+||∇ωtt||2)dt. $

    Applying Lemma 3.2 and the discrete Gronwall's inequality, we can complete the proof. □

    4.   New-CN-E-SAV scheme

    For the system (2.18)–(2.24), we obtain the semi-discrete new Crank−Nicolson E-SAV (New-CN-E-SAV) scheme by adopting the Crank−Nicolson scheme as

    $ ωn+1+ωn2=1Δt(yn+1−yn), $ (4.1)
    $ 1Δt(ωn+1−ωn)−Δyn+1+yn2−Q(ξn+1)f(ˆyn+12)=0, $ (4.2)
    $ ξn+1=exp(Rn+1)exp(H(ˆyn+12)), $ (4.3)
    $ Q(ξn+1)=ξn+1(2−ξn+1), $ (4.4)
    $ 1Δt(Rn+1−Rn)=0, $ (4.5)

    where $ \hat{y} ^{n+\frac{1}{2}} $ is obtained by solving the following equation:

    $ ˆwn+12=1Δt/2(ˆyn+12−yn), $ (4.6)
    $ 1Δt/2(ˆωn+12−ωn)−Δˆyn+12−f(yn)=0. $ (4.7)

    From (4.5), we can find that the New-CN-E-SAV scheme (4.1)–(4.5) also enjoys the same conservation as the CN-E-SAV scheme (3.1)–(3.3).

    Theorem 4.1. The New-CN-E-SAV scheme (4.1)–(4.5) is Hamiltonian conserved in the sense that

    $ Rn+1=Rn=⋅⋅⋅=R0. $

    Further, according to Theorem 4.1, the Eq (4.3) can be simplified as

    $ ξn+1=exp(R0)exp(H(ˆyn+12)). $ (4.8)

    In order to better illustrate the calculation process of the New-CN-E-SAV scheme (4.1)–(4.5), plugging (4.1) and (4.6) into (4.2) and (4.7), respectively, we obtain

    $ (2Δt2−12Δ)yn+1=2Δt2yn+2Δtωn+12Δyn+Q(ξn+1)f(ˆyn+12), $ (4.9)
    $ (4Δt2−Δ)ˆyn+12=4Δt2yn+2Δtωn+f(yn). $ (4.10)

    So the New-CN-E-SAV scheme can be implemented as follows:

    $ (i). solveˆyn+12from(4.10);(ii). solveˆωn+12from(4.6);(iii). computeξn+1from(4.8);(iv). computeQ(ξn+1)from(4.4);(v). solveyn+12from(4.9);(vi). solveωn+12from(4.1). $

    For the convergence order of the New-CN-E-SAV scheme, from (4.6) and (4.7), we can obtain

    $ ˆyn+12=y(tn+12)+O(Δt2), $ (4.11)
    $ ˆωn+12=ω(tn+12)+O(Δt2). $ (4.12)

    Referring to [46] and combining (4.11) and (4.12), we obtain

    $ ξn+1=exp(R0)exp(H(ˆyn+12))=ξ(tn+1)+CΔt=1+CΔt. $

    Then, we derive that

    $ Q(ξn+1)=ξn+1(2−ξn+1)=(ξ(tn+1)+CΔt)(2−ξ(tn+1)−CΔt)=(1+CΔt)(1−CΔt)=1−C2Δt2, $

    which means that the New-CN-E-SAV scheme can achieve second-order approximation.

    5.   Numerical experiments

    In this section, two examples are presented to test the validity of the theory. We set

    $ \Omega = [0,L]^{2},\; L = 2,\; T = 1, $
    $ y_{0} = 0.005 \sin \pi x_{1}\sin \pi x_{2}, $
    $ \omega_{0} = 0.005 \sin \pi x_{1}\sin \pi x_{2}, $

    and consider different nonlinear functions $ F(y) $ to simulate the order of convergence and the Hamiltonian conservation. Discretize the physical space by the Fourier spectral method with a spatial step $ h = L/2^9 $. Since we do not have the exact solution, we thus select the sufficiently small time step $ \Delta t = 1/1024 $ as the reference solution.

    Example 5.1. In this example, we choose the nonlinear function $ F(y) $ and initial values $ r_{0}, \mathcal{R}_{0} $ as follows:

    $ F(y) = 1-\cos y,\quad r_{0} = \int_{\Omega}F(y_{0})d\boldsymbol{x},\quad \mathcal{R}_{0} = H(y_{0}). $

    Example 5.2 In this example, we select

    $ F(y) = \frac{1}{4}(y^{2}-1)^{2},\quad r_{0} = \int_{\Omega}F(y_{0})d\boldsymbol{x},\quad \mathcal{R}_{0} = H(y_{0}). $

    For the CN-E-SAV and New-CN-E-SAV schemes, the error between the numerical solution and the exact solution in the sense of $ L^{2} $-norm is listed in Tables 1–6, where Tables 1–3 and 4–6 show the numerical results for Examples 5.1 and 5.2, respectively.

    Table 1.  Error results and convergence rate of $ \| e_{y}^{n}\| $ for the CN-E-SAV and New-CN-E-SAV schemes in Example 5.1.
    $ \Delta t $ $ \| e_{y}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 0.0368091884426225 -
    $ 1/2^{5} $ 0.00932236109731148 1.98129864079567
    $ 1/2^{6} $ 0.00233174733113645 1.99928392995304
    $ 1/2^{7} $ 0.000576576259379840 2.01582812615784
    $ 1/2^{8} $ 0.000137310038533674 2.07007433221585
    $ \| e_{y}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ New-CN-E-SAV } $ $ 1/2^{4} $ 0.0368091815051726 -
    $ 1/2^{5} $ 0.00932235922830067 1.98129865813145
    $ 1/2^{6} $ 0.00233174685298024 1.99928393655572
    $ 1/2^{7} $ 0.000576576140697115 2.01582812727880
    $ 1/2^{8} $ 0.000137310011085410 2.07007432364543
     | Show Table
    DownLoad: CSV
    Table 2.  Error results and convergence rate of $ \| e_{\omega}^{n}\| $ for the CN-E-SAV and New-CN-E-SAV schemes in Example 5.1.
    $ \Delta t $ $ \| e_{\omega}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 0.0689301039227398 -
    $ 1/2^{5} $ 0.0168861504409056 2.02929371796555
    $ 1/2^{6} $ 0.00418704131563571 2.01183741338057
    $ 1/2^{7} $ 0.00103299161999948 2.01910260335564
    $ 1/2^{8} $ 0.000245854829212462 2.07094995184852
    $ \| e_{\omega}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ New-CN-E-SAV } $ $ 1/2^{4} $ 0.0689300760782284 -
    $ 1/2^{5} $ 0.0168861432622035 2.02929374850860
    $ 1/2^{6} $ 0.00418703950543584 2.01183742378274
    $ 1/2^{7} $ 0.0010329911717826 2.01910260561751
    $ 1/2^{8} $ 0.000245854724513483 2.07094994024234
     | Show Table
    DownLoad: CSV
    Table 3.  Error results and convergence rate of $ | e_{r}^{n}| $ for the CN-E-SAV scheme in Example 5.1.
    $ \Delta t $ $ | e_{r}^{n}| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 2.86566749793482e-07 -
    $ 1/2^{5} $ 6.97595495640488e-08 2.03840859427113
    $ 1/2^{6} $ 1.72124441615129e-08 2.01893875053861
    $ 1/2^{7} $ 4.23455884358817e-09 2.02316839164966
    $ 1/2^{8} $ 1.00635793461970e-09 2.07306815290817
     | Show Table
    DownLoad: CSV
    Table 4.  Error results and convergence rate of $ \| e_{y}^{n}\| $ for the CN-E-SAV and New-CN-E-SAV schemes in Example 5.2.
    $ \Delta t $ $ \| e_{y}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 0.0371725171404962 -
    $ 1/2^{5} $ 0.00943514340320470 1.97812003334643
    $ 1/2^{6} $ 0.00236120638866597 1.99852029490610
    $ 1/2^{7} $ 0.000583930678977017 2.01565513477143
    $ 1/2^{8} $ 0.000139064798763014 2.07003983134903
    $ \| e_{y}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ New-CN-E-SAV } $ $ 1/2^{4} $ 0.0371725163498753 -
    $ 1/2^{5} $ 0.00943514322606903 1.97812002974700
    $ 1/2^{6} $ 0.00236120634599168 1.99852029389485
    $ 1/2^{7} $ 0.000583930668215755 2.01565513528492
    $ 1/2^{8} $ 0.000139064795893768 2.07003983452791
     | Show Table
    DownLoad: CSV
    Table 5.  Error results and convergence rate of $ \| e_{\omega}^{n}\| $ for the CN-E-SAV and New-CN-E-SAV schemes in Example 5.2.
    $ \Delta t $ $ \| e_{\omega}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 0.0818241862777186 -
    $ 1/2^{5} $ 0.0201549897717315 2.02139029902935
    $ 1/2^{6} $ 0.00500392653834192 2.01000453619864
    $ 1/2^{7} $ 0.00123483574539119 2.01874145867249
    $ 1/2^{8} $ 0.000293902813805396 2.07090807463406
    $ \| e_{\omega}^{n}\| $ $ \textbf{Rate} $
    $ \textbf{ New-CN-E-SAV } $ $ 1/2^{4} $ 0.0818241821912869 -
    $ 1/2^{5} $ 0.0201549887134675 2.02139030272942
    $ 1/2^{6} $ 0.00500392626767036 2.01000453848608
    $ 1/2^{7} $ 0.00123483567363466 2.01874146446974
    $ 1/2^{8} $ 0.000293902792746018 2.07090809417398
     | Show Table
    DownLoad: CSV
    Table 6.  Error results and convergence rate of $ | e_{r}^{n}| $ for the CN-E-SAV scheme in Example 5.2.
    $ \Delta t $ $ | e_{r}^{n}| $ $ \textbf{Rate} $
    $ \textbf{ CN-E-SAV } $ $ 1/2^{4} $ 3.36996122007882e-07 -
    $ 1/2^{5} $ 8.30406444718435e-08 2.02084244415644
    $ 1/2^{6} $ 2.05944534759084e-08 2.01156179910102
    $ 1/2^{7} $ 5.07805131277905e-09 2.01990896293877
    $ 1/2^{8} $ 1.20806775694149e-09 2.07157359980540
     | Show Table
    DownLoad: CSV

    The evolution of the Hamiltonian function with respect to the CN-E-SAV and New-CN-E-SAV schemes in Examples 5.1 and 5.2 is depicted in Figure 1.

    Figure 1.  Evolutions of the Hamiltonian for the CN-E-SAV and New-CN-E-SAV schemes with the time step $ \Delta t = \frac{1}{2^{6}} $ in Example 5.1 (left) and in Example 5.2 (right).
    DownLoad: Full-Size Img PowerPoint

    From the analysis of the data presented in Tables 1–6, it is evident that the error decreases as the time step $ \Delta t $ decreases. And the numerical results for the convergence order of the variables $ \omega, y, r $ in Examples 5.1 and 5.2 are consistent with the theoretical results. This thereby further illustrates the effectiveness of the CN-E-SAV and New-CN-E-SAV schemes.

    Figure 1 clearly shows that, regardless of the different nonlinear functions selected in the two examples, the CN-E-SAV and New-CN-E-SAV schemes successfully maintain the conservation property of the Hamiltonian function.

    6.   Conclusions

    In this work, we develop the CN-E-SAV and New-CN-E-SAV schemes to approximate a semi-linear wave equation with the following advantages: (ⅰ) it preserves Hamiltonian conservation; (ⅱ) it is efficient; and (ⅳ) it is easy to implement. The further topic can also involve designing the high-order BDFk-E-SAV schemes or applying them to other nonlinear problems. It is also meaningful to consider the construction of numerical schemes for the nonlinear localized wave equations.

    Use of AI tools declaration

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Acknowledgments

    This work is supported by the National Natural Science Foundation of China (Granted No. 11961008) and Guizhou University Doctoral Foundation (Granted NO. 15 (2022)).

    Conflict of interest

    The authors declare there is no conflict of interest.

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Article outline
Abstract
   Introduction
   Preliminary results
   CN-E-SAV scheme
   New-CN-E-SAV scheme
   Numerical experiments
   Conclusions
Use of AI tools declaration
Acknowledgments
Conflict of interest
References

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