
Diabetes is a metabolic disorder caused by insufficient insulin secretion and insulin secretion disorders. From health to diabetes, there are generally three stages: health, pre-diabetes and type 2 diabetes. Early diagnosis of diabetes is the most effective way to prevent and control diabetes and its complications. In this work, we collected the physical examination data from Beijing Physical Examination Center from January 2006 to December 2017, and divided the population into three groups according to the WHO (1999) Diabetes Diagnostic Standards: normal fasting plasma glucose (NFG) (FPG < 6.1 mmol/L), mildly impaired fasting plasma glucose (IFG) (6.1 mmol/L ≤ FPG < 7.0 mmol/L) and type 2 diabetes (T2DM) (FPG > 7.0 mmol/L). Finally, we obtained1,221,598 NFG samples, 285,965 IFG samples and 387,076 T2DM samples, with a total of 15 physical examination indexes. Furthermore, taking eXtreme Gradient Boosting (XGBoost), random forest (RF), Logistic Regression (LR), and Fully connected neural network (FCN) as classifiers, four models were constructed to distinguish NFG, IFG and T2DM. The comparison results show that XGBoost has the best performance, with AUC (macro) of 0.7874 and AUC (micro) of 0.8633. In addition, based on the XGBoost classifier, three binary classification models were also established to discriminate NFG from IFG, NFG from T2DM, IFG from T2DM. On the independent dataset, the AUCs were 0.7808, 0.8687, 0.7067, respectively. Finally, we analyzed the importance of the features and identified the risk factors associated with diabetes.
Citation: Yu-Mei Han, Hui Yang, Qin-Lai Huang, Zi-Jie Sun, Ming-Liang Li, Jing-Bo Zhang, Ke-Jun Deng, Shuo Chen, Hao Lin. Risk prediction of diabetes and pre-diabetes based on physical examination data[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3597-3608. doi: 10.3934/mbe.2022166
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Diabetes is a metabolic disorder caused by insufficient insulin secretion and insulin secretion disorders. From health to diabetes, there are generally three stages: health, pre-diabetes and type 2 diabetes. Early diagnosis of diabetes is the most effective way to prevent and control diabetes and its complications. In this work, we collected the physical examination data from Beijing Physical Examination Center from January 2006 to December 2017, and divided the population into three groups according to the WHO (1999) Diabetes Diagnostic Standards: normal fasting plasma glucose (NFG) (FPG < 6.1 mmol/L), mildly impaired fasting plasma glucose (IFG) (6.1 mmol/L ≤ FPG < 7.0 mmol/L) and type 2 diabetes (T2DM) (FPG > 7.0 mmol/L). Finally, we obtained1,221,598 NFG samples, 285,965 IFG samples and 387,076 T2DM samples, with a total of 15 physical examination indexes. Furthermore, taking eXtreme Gradient Boosting (XGBoost), random forest (RF), Logistic Regression (LR), and Fully connected neural network (FCN) as classifiers, four models were constructed to distinguish NFG, IFG and T2DM. The comparison results show that XGBoost has the best performance, with AUC (macro) of 0.7874 and AUC (micro) of 0.8633. In addition, based on the XGBoost classifier, three binary classification models were also established to discriminate NFG from IFG, NFG from T2DM, IFG from T2DM. On the independent dataset, the AUCs were 0.7808, 0.8687, 0.7067, respectively. Finally, we analyzed the importance of the features and identified the risk factors associated with diabetes.
The propagation of electromagnetic waves in some materials is usually modeled by the famous Maxwell's equations [4] with various proper medium responses. These significant responses reflect the material's properties, such as the magnetic permeability and electric permittivity with respect to the location and the frequency of the propagating field. When it turns to high intensity radiation situation, not only the medium quantities may depend on the magnitude of the propagating field, but also the response will become nonlinear. In nonlinear optics, one may often focus on the propagation of monochromatic waves, such as continuous high intensity laser beams. In this case, some reasonable assumptions (see [3]) simplify the Maxwell's models to a nonlinear Helmholtz (NLH) equation [5,13]
ΔE(X)+ω20c2n2(X,|E|)E(X)=f(X),n2(X,|E|)=n20(X)+2n0(X)n2(X)|E|2, | (1) |
where
Let the Kerr medium be surrounded by the linear homogeneous medium in which
v(X)=[n0(X)next0]2,ε(X)=2n2(X)n0(X)(next0)2. |
Then, equation (1) can be rewritten as
ΔE(X)+k20[v(X)+ε(X)|E(X)|2]E(X)=f(X). | (2) |
Since the Kerr medium coefficient
v={1,z<0,vint ,0⩽z⩽Zmax,1,z>Zmax,ε={0,z<0,εint ,0⩽z⩽Zmax,0,z>Zmax. |
Remark 1. When the Kerr medium is inhomogeneous which means the whole Kerr medium will be cut into pieces by some other mediums, the linear medium for example. In this case, the discontinuities of the coefficients
When the electric field
d2Edz2+k20(v+ε|E|2)E=f,z∈(0,Zmax). | (3) |
To solve (3) in the interval
E(0+)=E(0−),dEdz(0+)=dEdz(0−),E(Z+max)=E(Z−max),dEdz(Z+max)=dEdz(Z−max). |
According to this fact, the so-called two-way boundary conditions which are also used in [3,14,15] are as follows
(ddz+ik0)E|z=0=2ik0,(ddz−ik0)E|z=Zmax=0, | (4) |
where
Remark 2. According to [3], the above boundary condition (4) is developed from the inhomogeneous Sommerfeld type relation and its complete form is
(ddz+ik0)E|z=0=2ik0E0inc,(ddz−ik0)E|z=Zmax=−2ik0EZmaxinc, |
where
In 2D case, assuming the linear homogeneous medium in
ΔE+k20(v+ε|E|2)E=f,(x,y)∈Ω:=(ax,bx)×(ay,by),∂E∂n+ik0E=g,(x,y)∈Γ:=Γ1∪Γ2∪Γ3∪Γ4, | (5) |
where
Many researches are done for the NLH equation. Through transforming it into a phase-amplitude equation, Chen and Mills proposed an approach to obtain the closed form solution of the NLH equation with a single nonlinear layer [7] and multilayered structures [8]. And by using the multidimensional generalization of the nonlinear Schrödinger equation, the exact solution of the NLH equation in special case was also considered in [18]. Moreover, in [11,12], the existence and asymptotic behavior of the real-valued standing wave solution of the NLH equations were analyzed. On the other hand, numerical methods for the NLH equations are also investigated. Fibich and the collaborators studied the NLH equation in [1,2,3,14,15]: In [14], the authors constructed a two-way artificial boundary condition for the NLH equation to ensure that not only the backscattered waves generate no reflection but also the correct value of the incoming wave can be imposed; the NLH equation was also solved by using nonorthogonal expansions in [15]; by coupling with a new technique of the separation of variables, a fourth order finite difference scheme was developed in [1], and the algorithm was also extended to the three-dimensional axially symmetric problem; in [2,3], the authors solved the NLH equations by developing an efficient Newton's iteration method to deal with the strong nonlinearity. In addition, a finite element method which can approximate the discontinuous coefficient problem is constructed in [22,23]. Recently, Wu and Zou proved the existence and uniqueness of the NLH equation, and also analyzed the stability and the error estimate with explicit wave numbers for the finite element approximation in [29]. And the author proposed a robust modified Newton's method in [31].
There are many difficulties when the NLH equation is approximated by using numerical schemes. Firstly, since the NLH equation is a strong nonlinear problem, we need to search a robust iteration method for solving it. Secondly, similar to the Helmholtz equation, the solution of this problem is highly oscillating with a high number (see [19]). Moreover, it usually contains discontinuous coefficients due to the different propagating mediums. There are also other issues to be solved, such as the strong indefinite linear system generated from this equation and so on. But, in this paper, we mainly focus on the case that it admits highly oscillated solutions. It is well-known that, the FDM is one of the most frequently used numerical methods due to its simple structure. Usually, the FDM is constructed through approximating the derivative terms in the original equation with some difference quotients which are obtained by the Taylor's expansion directly [21,32,33], this may lead the FDM to suffer from some disadvantages, low computation accuracy for example. To simulate the Helmholtz equation with high wave numbers in one dimension, a new finite difference scheme was proposed in [25]. Being different from the classical one directly based on the Taylor expansion, the new finite difference method is constructed by a rearranged formula which can contain more regularity information of the solution, and thus more accurate approximations were achieved. Recently, this kind of schemes were applied to the higher dimensional problems in [16,17,26,27,28]. But all of these problems investigated above are linear problems. In this article, we will extend this idea to solve the NLH equation. Through some iteration methods, the NLH equation will be linearized as a linear one at each iterative step firstly. Several iteration methods are considered, including the classical ones and the error correction iteration method [24] in which the original iteration solution was modified by a residual. Then, based on the above resulted linear problem, the new finite difference scheme is constructed, which is naturally suitable for the problem with discontinuous coefficients to match the different propagating mediums (Kerr and linear mediums).
The rest of this paper is organized as follows. In the next section, we apply several iteration methods including the error correction one to linearize the NLH equations. Then, in Section 3, after constructing the new finite difference scheme for the 1D linearized equation, we extend the scheme to the 2D problem by using the ADI technique. To test the efficiency of the numerical schemes, some numerical experiments are performed in Section 4. We finally make conclusions in Section 5.
To solve the nonlinear equation, an iteration method is needed. In this section, we introduce several kinds of iteration methods for solving the NLH equation. For convenience, we rewrite the problem as,
LE+k20ε|E|2E=f, | (6) |
where
Frozen-nonlinearity iteration may be the simplest iteration method in which the nonlinear term is frozen as a known quantity. For example, by replacing
LEl+1+k20ε|El|2El+1=f, | (7) |
where
λ(E,¯E):=LE+k20εE2¯E−f=0, | (8) |
then there hold,
∂λ∂E(E,¯E)=L+2k20εE¯E,∂λ∂¯E(E,¯E)=k20εE2, | (9) |
where
λ(El,¯El)+∂λ∂E(El,¯El)s+∂λ∂¯E(El,¯El)ˉs=0. | (10) |
Letting
LEl+1+2k20ε|El|2El+1+k20ε(El)2¯El+1=f+2k20ε|El|2El. | (11) |
Furthermore, a modified Newton's method was proposed in [31] by replacing
LEl+1+2k20ε|El|2El+1=f+k20ε|El|2El. | (12) |
In this paper, we also employ the error correction method in [24] for solving the nonlinear Helmholtz equation. Next, we will show the process. For simplicity, the above three iteration methods can be rewritten as a general formula
L˜El+1+N˜El+1+M¯˜El+1=˜f, | (13) |
where
N:={k20ε|El|2,in (7),2k20ε|El|2,in (11),2k20ε|El|2,in (12),M:={0,in (7),k20ε(El)2,in (11),0,in (12),˜f:={f,in (7),f+2k20ε|El|2El,in (11),f+k20ε|El|2El,in (12). |
Assuming that
Lμl+1+k20ε|E|2E−N˜El+1−M¯˜El+1=f−˜f. | (14) |
Since
|E|2E=|˜El+1+μl+1|2(˜El+1+μl+1)=(2|˜El+1|2+|μl+1|2)μl+1+(˜El+1)2¯μl+1+|˜El+1|2˜El+1+2|μl+1|2˜El+1+(μl+1)2¯˜El+1, |
freezing the terms
Lμl+1+Pμl+1+Q¯μl+1=fμ, | (15) |
where
P=2k20ε|˜El+1|2+k20ε|μl|2,Q=k20ε(˜El+1)2,fμ=f−˜f−(2k20ε|μl|2+k20ε|˜El+1|2−N)˜El+1−[k20ε(μl)2−M]¯˜El+1. |
Furthermore, for the error on the boundary, there holds
∂μl+1∂n+ik0μl+1=0. | (16) |
Obviously, by solving (15), the original solution
Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
After applying the iteration methods introduced in the above section, the NLH equation is linearized to linear problems at each iterative step. To implement these methods, a spatial discretization scheme is needed. From the stability analysis in [29], we know that the solution of the NLH equation satisfying
Recalling the iteration formula for the 1D NLH equation
d2˜El+1dz2+(k20v+N)˜El+1+M¯˜El+1=˜f,z∈(0,Zmax), | (17) |
(ddz+ik0)˜El+1|z=0=2ik0,(ddz−ik0)˜El+1|z=Zmax=0. | (18) |
When the frozen-nonlinearity iteration (7) and the modified Newton's method (12) are used, i.e.,
d2˜El+1dz2=τ˜El+1+˜f, | (19) |
where
Since the parameters
τm={τ−m,zm∈[zm−1,zm],τ+m,zm∈[zm,zm+1],˜fm={˜f−m,zm∈[zm−1,zm],˜f+m,zm∈[zm,zm+1]. |
Taylor's expansion tells that
˜El+1m+1=˜El+1m+⋯+(h+m)2k(2k)!(˜El+1m)(2k)+(h+m)2k+1(2k+1)!(˜El+1m)(2k+1)+⋯, | (20) |
˜El+1m−1=˜El+1m+⋯+(h−m)2k(2k)!(˜El+1m)(2k)−(h−m)2k+1(2k+1)!(˜El+1m)(2k+1)+⋯. | (21) |
And according to (19), there hold
(˜El+1m)(2k)=τnm˜El+1m+k∑s=1τk−sm˜f(2s−2)m, | (22) |
(˜El+1m)(2k+1)=τnm(˜El+1m)(1)+k∑s=1τk−sm˜f(2s−1)m. | (23) |
Substituting (22)-(23) into (20) and (21), it yields
˜El+1m+1=G+m˜El+1m+H+m(˜El+1m)(1)++∞∑k=1[L+k;m(˜f+m)(2k−2)+X+k;m(˜f+m)(2k−1)], | (24) |
˜El+1m−1=G−m˜El+1m−H−m(˜El+1m)(1)++∞∑k=1[L−k;m(˜f−m)(2k−2)−X−k;m(˜f−m)(2k−1)], | (25) |
where
G±m=G(τ±m,h±m),H±m=H(τ±m,h±m),L±k;m=Lk(τ±m,h±m),X±k;m=Xk(τ±m,h±m), |
with
G(ρ,h):=eh√ρ+e−h√ρ2,H(ρ,h):=1√ρeh√ρ−e−h√ρ2,Lk(ρ,h):=1ρk[eh√ρ+e−h√ρ2−k−1∑s=0(h√ρ)2s(2s)!],Xk(ρ,h):=1ρk+1/2[eh√ρ−e−h√ρ2−k−1∑s=0(h√ρ)2s+1(2s+1)!]. |
Then, eliminating
H+m˜El+1m−1−(H+mG−m+H−mG+m)˜El+1m+H−m˜El+1m+1=+H−m+∞∑k=1[L+k;m(˜f+m)(2k−2)+X+k;m(˜f+m)(2k−1)]+H+m+∞∑k=1[L−k;m(˜f−m)(2k−2)−X−k;m(˜f−m)(2k−1)]. | (26) |
For the boundary points
˜El+12=G+1˜El+11+H+1(˜El+11)(1)++∞∑k=1[L+k;1(˜f+1)(2k−2)+X+k;1(˜f+1)(2k−1)],˜El+1N−1=G−N˜El+1N−H−N(˜El+1N)(1)++∞∑k=1[L−k;N(˜f−N)(2k−2)−X−k;N(˜f−N)(2k−1)]. |
Then, substituting the boundary condition (18) into the above formulas, we have the numerical schemes for
−(G+1−ik0H+1)˜El+11+˜El+12=2ik0H+1++∞∑k=1[L+k;1(˜f+1)(2k−2)+X+k;1(˜f+1)(2k−1)], | (27) |
˜El+1N−1−(G−N−ik0H−N)˜El+1N=+∞∑k=1[L−k;N(˜f−N)(2k−2)−X−k;N(˜f−N)(2k−1)]. | (28) |
Obviously, taking different
H+m˜El+1m−1−(H+mG−m+H−mG+m)˜El+1m+H−m˜El+1m+1=+H−m[L+1;m˜f+m+2X+1;mh−m+h+m(˜f+m+1−˜f+m−1)]+H+m[L−1;m˜f−m−2X−1;mh−m+h+m(˜f−m+1−˜f−m−1)],m=2,3,⋯,N−1,−(G+1−ik0H+1)˜El+11+˜El+12=2ik0H+1+L+1;1˜f+1+X+1;1h+1(˜f+2−˜f+1), | (29) |
˜El+1N−1−(G−N−ik0H−N)˜El+1N=L−1;N˜f−N−X−1;Nh−N(˜f−N−˜f−N−1). |
Remark 3. In fact, a more accurate numerical scheme could be developed. For example, when the frozen-nonlinearity iteration method is used, (19) is the equation with variable coefficient, it yields,
d2˜El+1dz2=τ(z)˜El+1+˜f, |
where
In this case, like (22)-(23), we can get more precise formulas
(˜El+1m)(2n)=τnm˜El+1m+τn−1m˜fm,(˜El+1m)(2n+1)=τnm(˜El+1m)(1)+(C01+C23+⋯+C2n−22n−1)τn−1mτ(1)m˜El+1m+τn−1m˜f(1)m, |
where
Then, according to the Taylor's series, we get
˜El+1m=A+m˜Em+B+m˜E(1)m+C+m˜fm+D+m˜f(1)m,˜El−1m=A−m˜Em+B−m˜E(1)m+C−m˜fm+D−m˜f(1)m, |
where
A±m=A(τ±m,h±m),B±m=−B(τ±m,h±m),C±m=C(τ±m,h±m),D±m=−D(τ±m,h±m),A(ν,h)=12[e√νh+e−√νh]+ν(1)8{1(√ν)3[e√νh−e−√νh−2√νh]}+ν(1)8{−h(√ν)2[e√νh+e−√νh−2]+h2√ν[e√νh−e−√νh]},B(ν,h)=12√ν[e√νh−e−√νh],C(ν,h)=12ν[e√νh+e−√νh],D(ν,h)=12ν√ν[e√νh−e−√νh]. |
Thus, eliminating the terms
−1B−m˜Em−1−(A+mB+m−A−mB−m)˜Em+1B+m˜Em+1=(C+mB+m−C−mB−m)˜fm+(D+mB+m−D−mB−m)˜f(1)m. | (30) |
Finally, by approximating
When the Newton's iteration method (11) is considered, (17) needs to be separated into the real and imaginary parts due to the existence of
d2Rdz2=ˆRR+ˆII+˜fR, | (31) |
(dRdz−k0I)|z=0=0,(dRdz+k0I)|z=Zmax=0, | (32) |
d2Idz2=⌢II+⌢RR+˜fI, | (33) |
(dIdz+k0R)|z=0=2k0,(dIdz−k0R)|z=Zmax=0, | (34) |
where
R=real(˜El+1),I=imag(˜El+1),˜fR=real(˜f),˜fI=imag(˜f),ˆR=−real(k20v+N+M),ˆI=−imag(M−k20v−N),⌢I=−real(k20v+N−M),⌢R=−imag(M+k20v+N). |
Taking the real part equation (31) for an example, following the same process for (19), we have, at any interior point
R(2k)m=ˆRkmRm+ˆImk∑s=1ˆRk−smI(2s−2)m+k∑s=1ˆRk−sm˜f(2s−2)R;m, | (35) |
R(2k+1)m=ˆRkmR(1)m+ˆImk∑s=1ˆRk−smI(2s−1)m+k∑s=1ˆRk−sm˜f(2s−1)R;m, | (36) |
and
Rm+1=G+mRm+H+mR(1)m++∞∑k=1[L+k;m(ˆI+mIm+˜f+R;m)(2k−2)+X+k;m(ˆI+mIm+˜f+R;m)(2k−1)], | (37) |
Rm−1=G−mRm−H−mR(1)m++∞∑k=1[L−k;m(ˆI−mIm+˜f−R;m)(2k−2)−X−k;m(ˆI−mIm+˜f−R;m)(2k−1)], | (38) |
where
G±m=G(ˆR±m,h±m),H±m=H(ˆR±m,h±m),L±k;m=Lk(ˆR±m,h±m),X±k;m=Xk(ˆR±m,h±m). |
Eliminating
H+mRm−1−(H+mG−+H−mG+)Rm+H−mRm+1=+∞∑k=1(H−mF+k+H+mF−k), | (39) |
where
F+k=ˆI+m(L+k;mI(2k−2)m+X+k;mI(2k−1)m)+L+k;m(˜f+R;m)(2k−2)+X+k;m(˜f+R;m)(2k−1)F−k=ˆI−m(L−k;mI(2k−2)m−X−k;mI(2k−1)m)+L−k;m(˜f−R;m)(2k−2)−X+k;m(˜f−R;m)(2k−1). |
Similarly, setting
−G+1R1+R2−k0H+1I1=+∞∑k=1[ˆI+1(L+k;1I(2k−2)1+X+k;1I(2k−1)1)+L+k;1(˜f+R;1)(2k−2)+X+k;1(˜f+R;1)(2k−1)], | (40) |
RN−1−G−NRN−k0H−NIN=+∞∑k=1[ˆI−N(L−k;NI(2k−2)N−X−k;NI(2k−1)N)+L−k;N(˜f−R;N)(2k−2)−X+k;N(˜f−R;N)(2k−1)]. | (41) |
Obviously, by retaining different terms in the right hand side of (39)-(41), we can also get a series of finite difference schemes for the real part equation (31)-(32). For example, taking
H+mRm−1−(H+mG−+H−mG+)Rm+H−mRm+1=(H−mˆI+mL+1;m+H+mˆI−mL−1;m)Im+(H−mˆI+mX+1;m−H+mˆI−mX−1;m)I(1)m+H−mL+1;m(˜f+R;m)+H+mL−1;m(˜f−R;m)H−mX+1;m(˜f+R;m)(1)−H+mX−1;m(˜f−R;m)(1). |
By approximating
A⋅[R,I]=B⋅FR, | (42) |
where
A=(A1,A2,A3,A4,A5,A6),R=(Rm−1,Rm,Rm+1),I=(Im−1,Im,Im+1),B=(B1,B2,B3,B4,B5,B6),FR=(˜f+R;m−1,˜f+R;m,˜f+R;m+1,˜f−R;m−1,˜f−R;m,˜f−R;m+1), |
with
A1=H+m,A2=−H+mG−m−H−mG+m,A3=H−m,A4=−A6=H−mˆI+mX+1;m−H+mˆI−mX−1;mh+m+h−m,A5=−H−mˆI+mL+1;m−H+mˆI−mL−1;m,B1=−B3=−H−mX+1;mh+m+h−m,B2=H−mL+1;m,B4=−B6=H+mX−1;mh+m+h−m,B5=H+mL−1;m. |
Similarly, letting
−G+1R1+R2−k0H+1I1=ˆI+1(L+1;1I1+X+1;1I(1)1)+L+1;1˜f+R;1+X+1;1(˜f+R;1)(1),RN−1−G−NRN−k0H−NIN=ˆI−N(L−1;NIN−X−1;NI(1)N)+L−1;N˜f−R;N−X+1;N(˜f−R;N)(1). |
By using the boundary condition (34) and approximating
−(G+1−k0ˆI+1X−1;1)R1+R2−(k0H+1+L+1;1ˆI+1)I1=(L+1;1−X+1;1h+1)˜f+R,1+X+1;1h+1˜f+R,2+2k0ˆI+1X−1;1, | (43) |
RN−1−(G−N−k0X−1;NˆI−N)RN−(k0H−N+L−1;NˆI−N)IN=(L−1;N−X−1;Nh−N)˜f−R;N+X−1;Nh−N˜f−R,N−1. | (44) |
For the imaginary part (33)-(34), the same procedure can be applied to develop the new finite difference scheme at any interior point
A⋅[I,R]=B⋅FI, | (45) |
where
FI=(˜f+I;m−1,˜f+I;m,˜f+I;m+1,˜f−I;m−1,˜f−I;m,˜f−I;m+1),A1=T+m,A2=−T+mS−m−T−mS+m,A3=T−m,A4=−A6=T−m⌢R+mY+1;m−T+m⌢R−mY−1;mh+m+h−m,A5=−T−m⌢R+mM+1;m−T+m⌢R−mM−1;m,B1=−B3=−T−mY+1;mh+m+h−m,B2=T−mM+1;m,B4=−B6=T+mY−1;mh+m+h−m,B5=T+mM−1;m, |
with
S±m=G(⌢I±m,h±m),T±m=H(⌢I±m,h±m),M±1;m=L1(⌢I±m,h±m),Y±1;m=X1(⌢I±m,h±m). |
And the schemes for boundary points
−(S+1+k0⌢R+1Y+1;1)I1+I2−(M+1;1⌢R+1−k0T+1)R1=(M+1;1−Y+1;1h+1)˜f+I,1+Y+1;1h+1˜f+I,2+2k0T+1, | (46) |
IN−1−(S−N+k0Y−1;N⌢R−N)IN−(M−1;N⌢R−N−k0T−N)RN=(M−1;N−Y−1;Nh−N)˜f−I;N+Y−1;Nh−N˜f−I,N−1. | (47) |
Obviously, (42)-(47) constitute a finite difference scheme for the system of equations (31)-(34). According to the above process, it can be found that, through translating the high order terms
In this section, we extend the new finite difference scheme to the 2D problem by applying the ADI method [9,10]. Similar to (31)-(34), the 2D equation (5) need to be divided into real and imaginary parts like
ΔR=ˆRR+ˆII+˜fR,(x,y)∈Ω, | (48) |
−∂R∂y−k0I=g1R,(x,y)∈Γ1,∂R∂y−k0I=g2R,(x,y)∈Γ2,−∂R∂x−k0I=g3R,(x,y)∈Γ3,∂R∂x−k0I=g4R,(x,y)∈Γ4, | (49) |
ΔI=⌢II+⌢RR+˜fI,(x,y)∈Ω, | (50) |
−∂I∂y+k0R=g1I,(x,y)∈Γ1,−∂I∂y+k0R=g2I,(x,y)∈Γ2,−∂I∂x+k0R=g3I,(x,y)∈Γ3,∂I∂x+k0R=g4I,(x,y)∈Γ4, | (51) |
where
According to (5) and Fig. 2, the parameters
It is well-known that the ADI method is used to simulate a high-dimensional problem by solving a series of one-dimensional problems. Based on this, by directly separating the real part equation (48) into two 1D equations in
∂2Rm,n∂x2=γxˆRm,nRm,n+γxˆIm,nIm,n+˜fxR;m,n, | (52) |
∂2Rm,n∂y2=γyˆRm,nRm,n+γyˆIm,nIm,n+˜fyR;m,n, | (53) |
where
Similar to (42), the new finite difference schemes for (52) and (53) can be directly got as follows
H+xRm−1,n−(H+xG−x+H−xG+x)Rm,n+H−xRm+1,n−(H−xγxˆI+m,nL+x,1+H+xγxˆI−m,nL−x,1)Im,n−H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1h+m+h−m(Im+1,n−Im−1,n)=(H−xL+x,1+H+xL−x,1)˜fxR;m,n,H+yRm,n−1−(H+yG−y+H−yG+y)Rm,n+H−yRm,n+1−(H−yγyˆI+m,nL+y,1+H+yγyˆI−m,nL−y,1)Im,n | (54) |
−H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1k+n+k−n(Im,n+1−Im,n−1)=(H−yL+y;1+H+yL−y;1)˜fyR;m,n, | (55) |
where
G±x=G(γxˆR±m,n,h±m),H±x=H(γxˆR±m,n,h±m),L±x,1=L1(γxˆR±m,n,h±m),X±x,1=X1(γxˆR±m,n,h±m),G±y=G(γyˆR±m,n,k±n),H±y=H(γyˆR±m,n,k±n),L±y;1=L1(γyˆR±m,n,k±n),X±y;1=X1(γyˆR±m,n,k±n). |
Combining (54) and (55), we get the new finite difference scheme for (48) at the interior point
A⋅[R,I]=˜fR;m,n, | (56) |
where
A=(A1,A2,A3,A4,A5,A6,A7,A8,A9,A10),R=(Rm,n−1,Rm−1,n,Rm,n,Rm+1,n,Rm,n+1),I=(Im,n−1,Im−1,n,Im,n,Im+1,n,Im,n+1), |
with
A1=H+yH−yL+y;1+H+yL−y;1,A2=H+xH−xL+x,1+H+xL−x,1,A3=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−H+yG−y+H−yG+yH−yL+y;1+H+yL−y;1,A4=H−xH−xL+x,1+H+xL−x,1,A5=H−yH−yL+y;1+H+yL−y;1,A6=H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1),A7=H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A8=−H−xγxˆI+m,nL+x,1+H+xγxˆI−m,nL−x,1H−xL+x,1+H+xL−x,1−H−yγyˆI+m,nL+y,1+H+yγyˆI−m,nL−y,1H−yL+y;1+H+yL−y;1,A9=−H−xγxˆI+m,nX+x,1+H+xγxˆI−m,nX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A10=−H−yγyˆI+m,nX+y,1+H+yγyˆI−m,nX−y,1(k+n+k−n)(H−yL+y;1+H+yL−y;1). |
Similar to the interior points, the new finite difference scheme for each boundary point is also constructed by developing two schemes in
Rm,2=G+yRm,1+H+y∂R(1)m,1∂y+L+y,1˜fyR;m,1+γyˆI+m,1(L+y;1Im,1+X+y,1∂I(1)m,1∂y). |
Then, substituting the corresponding boundary condition into the above formula, we have
−(G+y+k0γyˆI+m,1X+y,1)Rm,1+Rm,2+(k0H+y−γyˆI+m,1L+y,1)Im,1=L+y;1˜fyR;m,1−H+yg1R;m,1−γyˆI+m,1X+y;1g1I;m,1. | (57) |
So, combining (54)(
Ab⋅Rb=Fb, | (58) |
where
Rb=(Rm−1,1,Rm,1,Rm+1,1,Im−1,1,Im,1,Im+1,1,Rm,2),Ab=(A1,A2,A3,A4,A5,A6,A7),A1=H+xH−xL+x,1+H+xL−x,1,A3=H−xH−xL+x,1+H+xL−x,1,A2=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−G+y+k0γyˆI+m,1X+y,1L+y;1,A4=H−xγxˆI+m,1X+x,1+H+xγxˆI−m,1X−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A5=k0H+y−γyˆI+m,1L+y,1L+y;1−H−xγxˆI+m,1L+x,1+H+xγxˆI−m,1L−x,1H−xL+x,1+H+xL−x,1,A6=−H−xγxˆI+m,1X+x,1+H+xγxˆI−m,1X−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A7=1L+y;1,Fb=˜fR;m,1−H+yL+y;1g1R;m,1−γyˆI+m,1X+y;1L+y;1g1I;m,1. |
Similar to (57), on other three boundaries (excluding the vertexes), we also have: for the boundary points on
Rm,Ny−1−(G−y+k0γyˆI−m,NyX−y;1)Rm,Ny+(k0H−y−γyˆI−m,NyL−y;1)Im,Ny=L−y;1˜fyR;m,Ny−H−yg2R;m,Ny−γyˆI−m,NyX−y;1g2I;m,Ny, | (59) |
for the boundary points on
−(G+x+k0γxˆI+1,nX+x;1)R1,n+R2,n+(k0H+x−γxˆI+1,nL+x,1)I1,n=L+x;1˜fxR;1,n−H+xg3R;1,n−γxˆI+1,nX+x;1g3I;1,n, | (60) |
for the boundary points on
RNx−1,n−(G−x+k0γxˆINx,nX−x;1)RNx,n+(k0H−x−γxˆINx,nL−x,1)INx,n=L−x,1˜fxR;Nx,n−H−xg4R;Nx,n−γxˆI−Nx,nX−x;1g4I;Nx,n. | (61) |
Thus, applying the same process, the new finite difference schemes for these boundary points can also be written as (58) with different
Rb=(Rm−1,Ny,Rm,Ny,Rm+1,Ny,Im−1,Ny,Im,Ny,Im+1,Ny,Rm,Ny−1),A1=H+xH−xL+x,1+H+xL−x,1,A3=H−xH−xL+x,1+H+xL−x,1,A2=−H+xG−x+H−xG+xH−xL+x,1+H+xL−x,1−G−y+k0γyˆI−m,NyX−y;1L−y;1,A4=H−xγxˆI+m,NyX+x,1+H+xγxˆI−m,NyX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A5=k0H−y−γyˆI−m,NyL−y;1L−y;1−H−xγxˆI+m,NyL+x,1+H+xγxˆI−m,NyL−x,1H−xL+x,1+H+xL−x,1,A6=−H−xγxˆI+m,NyX+x,1+H+xγxˆI−m,NyX−x,1(h+m+h−m)(H−xL+x,1+H+xL−x,1),A7=1L−y;1,Fb=˜fR;m,Ny−H−yg2R;m,NyL−y;1−γyˆI−m,NyX−y;1g2I;m,NyL−y;1, |
and for the points on
\begin{align*} &\mathbf{R_b} = (R_{1,n-1},R_{1,n},R_{1,n+1},I_{1,n-1},I_{1,n},I_{1,n+1},R_{2,n}),\\ &A_1 = \frac{H_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\quad A_3 = \frac{H_{y}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\\ &A_2 = -\frac{G_{x}^{+}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{1, n}^{+} X_{x;1}^{+}}{L_{x ; 1}^{+}} -\frac{H_{y}^{+} G_{y}^{-}+H_{y}^{-} G_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} , \\ &A_4 = \frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})},\\ &A_5 = \frac{k_{0} H_{x}^{+}-\gamma_{x} \widehat{\mathcal{I}}_{1, n}^{+} L_{x, 1}^{+}}{L_{x ; 1}^{+}} -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{+} L_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{-} L_{y, 1}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}},\\ &A_6 = -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{1, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})}, \\ &A_7 = \frac{1}{L_{x ; 1}^{+}},\quad F_b = \widetilde{f}_{R ; 1, n}-\frac{H_{x}^{+} g_{3 R ; 1, n}}{L_{x ; 1}^{+}} -\frac{\gamma_{x} \widehat{\mathcal{I}}_{1, n}^{+} X_{x ; 1}^{+} g_{3 I ; 1, n}}{L_{x ; 1}^{+}}, \end{align*} |
and for the points on
\begin{align*} &\mathbf{R_b} = (R_{N_x,n-1},R_{N_x,n},R_{N_x,n+1},I_{N_x,n-1},I_{N_x,n},I_{N_x,n+1},R_{N_x-1,n}),\\ &A_1 = \frac{H_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\quad A_3 = \frac{H_{y}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} ,\\ &A_2 = -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n} X_{x;1}^{-}}{L_{x, 1}^{-}} -\frac{H_{y}^{+} G_{y}^{-}+H_{y}^{-} G_{y}^{+}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}} , \\ &A_4 = \frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})}, \end{align*} |
\begin{align*} &A_5 = \frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n} L_{x, 1}^{-}}{L_{x, 1}^{-}} -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} L_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} L_{y, 1}^{-}}{H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-}},\\ &A_6 = -\frac{H_{y}^{-} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{+} X_{y, 1}^{+}+H_{y}^{+} \gamma_{y} \widehat{\mathcal{I}}_{N_x, n}^{-} X_{y, 1}^{-}}{(k_{n}^{+}+k_{n}^{-})(H_{y}^{-} L_{y ; 1}^{+}+H_{y}^{+} L_{y ; 1}^{-})}, \\ &A_7 = \frac{1}{L_{x, 1}^{-}},\quad F_b = \widetilde{f}_{R ; N_{x}, n}-\frac{H_{x}^{-} g_{4 R ; N_{x}, n}}{L_{x, 1}^{-}} -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, n}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, n}}{L_{x, 1}^{-}}. \end{align*} |
According to (57), (59)-(61), the new finite schemes for four vertexes can also be obtained. For example setting
\begin{align} \mathbf{A_v\cdot R_v} = F_v, \end{align} | (62) |
where
\begin{align*} &\mathbf{A_v} = (A_1,A_2,A_3,A_4),\quad \mathbf{R_v} = (R_{1,1},R_{2,1},R_{1,2},I_{1,1}),\\ &A_1 = -\frac{G_{y}^{+}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} X_{y, 1}^{+}}{L_{y ; 1}^{+} } -\frac{G_{x}^{+}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} X_{x;1}^{+}}{L_{x ; 1}^{+} },\\ &A_2 = \frac{1}{L_{x ; 1}^{+} },\quad A_3 = \frac{1}{L_{y ; 1}^{+} },\quad A_4 = \frac{k_{0} H_{y}^{+}-\gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} L_{y, 1}^{+}}{L_{y ; 1}^{+} } +\frac{k_{0} H_{x}^{+}-\gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} L_{x, 1}^{+}}{L_{x ; 1}^{+} },\\ &F_v = \widetilde{f}_{R ; 1, 1}-\frac{H_{y}^{+} }{L_{y ; 1}^{+} }g_{1 R ; 1, 1} -\frac{\gamma_{y} \widehat{\mathcal{I}}_{1, 1}^{+} X_{y ; 1}^{+}}{L_{y ; 1}^{+} } g_{1 I ; 1, 1} -\frac{H_{x}^{+} g_{3 R ; 1, 1}}{L_{x ; 1}^{+} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{1, 1}^{+} X_{x ; 1}^{+} g_{3 I ; 1, 1}}{L_{x ; 1}^{+} }. \end{align*} |
Similarly, for the rest three vertexes, their new finite difference schemes can be also concluded in (62) with different
\begin{align*} &\mathbf{R_v} = (R_{N_x-1,1},R_{N_x,1},R_{N_x,2},I_{N_x,1}),\\ &A_1 = \frac{1}{L_{x, 1}^{-} },\quad A_3 = \frac{1}{L_{y ; 1}^{+} },\\ &A_2 = -\frac{G_{y}^{+}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} X_{y, 1}^{+}}{L_{y ; 1}^{+} } -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1} X_{x;1}^{-}}{L_{x, 1}^{-} } ,\\ &A_4 = \frac{k_{0} H_{y}^{+}-\gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} L_{y, 1}^{+}}{L_{y ; 1}^{+} } +\frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1} L_{x, 1}^{-}}{L_{x, 1}^{-} } ,\\ &F_v = \widetilde{f}_{R ; N_x, 1}-\frac{H_{y}^{+} }{L_{y ; 1}^{+} }g_{1 R ; N_x, 1} -\frac{\gamma_{y} \widehat{\mathcal{I}}_{N_x, 1}^{+} X_{y ; 1}^{+}}{L_{y ; 1}^{+} } g_{1 I ; N_x, 1}\\ &-\frac{H_{x}^{-} g_{4 R ; N_{x}, 1}}{L_{x, 1}^{-} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, 1}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, 1}}{L_{x, 1}^{-} }, \end{align*} |
and for the vertex
\begin{align*} &\mathbf{R_v} = (R_{1,N_y-1},R_{1,N_y},R_{2,N_y},I_{1,N_y}),\\ &A_1 = \frac{1}{L_{y ; 1}^{-} },\quad A_3 = \frac{1}{L_{x ; 1}^{+} }, \end{align*} |
\begin{align*} &A_2 = -\frac{G_{y}^{-}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} X_{y;1}^{-}}{L_{y ; 1}^{-} } -\frac{G_{x}^{+}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} X_{x;1}^{+}}{L_{x ; 1}^{+} },\\ &A_4 = \frac{k_{0} H_{y}^{-}-\gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} L_{y ; 1}^{-}}{L_{y ; 1}^{-} } \frac{k_{0} H_{x}^{+}-\gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} L_{x, 1}^{+}}{L_{x ; 1}^{+} } ,\\ &F_v = \widetilde{f}_{R ; 1, N_{y}}-\frac{H_{y}^{-} g_{2 R ; 1, N_{y}}}{L_{y ; 1}^{-} } -\frac{\gamma_{y} \widehat{\mathcal{I}}_{1, N_{y}}^{-} X_{y ; 1}^{-} g_{2 I ; 1, N_{y}}}{L_{y ; 1}^{-} }\\ &\qquad-\frac{H_{x}^{+} g_{3 R ; 1, N_y}}{L_{x ; 1}^{+} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{1, N_y}^{+} X_{x ; 1}^{+} g_{3 I ; 1, N_y}}{L_{x ; 1}^{+} }, \end{align*} |
and for the vertex
\begin{align*} &\mathbf{R_v} = (R_{N_x,N_y-1},R_{N_x-1,N_y},R_{N_x,N_y},I_{N_x,N_y}),\\ &A_1 = \frac{1}{L_{y ; 1}^{-} },\quad A_2 = \frac{1}{L_{x, 1}^{-} },\\ &A_3 = -\frac{G_{y}^{-}+k_{0} \gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} X_{y;1}^{-}}{L_{y ; 1}^{-} } -\frac{G_{x}^{-}+k_{0} \gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y} X_{x;1}^{-}}{L_{x, 1}^{-} } ,\\ &A_4 = \frac{k_{0} H_{y}^{-}-\gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} L_{y ; 1}^{-}}{L_{y ; 1}^{-} } +\frac{k_{0}H_{x}^{-}-\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y} L_{x, 1}^{-}}{L_{x, 1}^{-} },\\ &F_v = \widetilde{f}_{R ; N_x, N_{y}}-\frac{H_{y}^{-} g_{2 R ; N_x, N_{y}}}{L_{y ; 1}^{-} } -\frac{\gamma_{y} \widehat{\mathcal{I}}_{N_x, N_{y}}^{-} X_{y ; 1}^{-} g_{2 I ; N_x, N_{y}}}{L_{y ; 1}^{-} }\\ &\qquad-\frac{H_{x}^{-} g_{4 R ; N_{x}, N_y}}{L_{x, 1}^{-} } -\frac{\gamma_{x} \widehat{\mathcal{I}}_{N_{x}, N_y}^{-} X_{x ; 1}^{-} g_{4 I ; N_{x}, N_y}}{L_{x, 1}^{-} }. \end{align*} |
For the imaginary part (50)-(51), the new finite difference scheme can be produced in the same way. And the details are omitted here.
Remark 4. In fact, for the 2D equation, when the frozen-nonlinearity iteration and the modified Newton's method are used, it is not necessary to separate (5) into real and imaginary parts. In this case, the 2D equation can be divided into two 1D equations directly. Furthermore, since two 1D equations are separated from a 2D equation, we assume that
In this section, we will show some numerical tests to verify the efficiency of the scheme proposed in the above section. And we set
Firstly, let
\begin{align} E& = \lambda(z) e^{i \varphi(z)}, \end{align} | (63) |
where
\begin{align} \varphi(z)& = \varphi(0)+W \int_{0}^{z} \frac{1}{\beta(t)} d t, \end{align} | (64) |
\begin{align} \lambda(z)& = \sqrt{\beta(z)}, \end{align} | (65) |
with
\begin{align} \beta(z)& = \beta_{2}+(\beta_{1}-\beta_{2}) \operatorname{cn}^{2}\left[\sqrt{\frac{\varepsilon}{2v}}\left(\beta_{1}-\beta_{3}\right)^{1 / 2} k_0\sqrt{v}(Z_{\max}-z)\bigg | \frac{\beta_{1}-\beta_{2}}{\beta_{1}-\beta_{3}}\right], \end{align} | (66) |
\begin{align} \beta_{1}& = \frac{W}{k_{0}}, \end{align} | (67) |
\begin{align} \beta_{2}& = \frac{v}{\varepsilon}\left\{\left[\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)^{2}+\frac{2 \varepsilon W}{v^{2} k_{0}}\right]^{1 / 2}-\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)\right\}, \end{align} | (68) |
\begin{align} \beta_{3}& = -\frac{v}{\varepsilon}\left\{\left[\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)^{2}+\frac{2 \varepsilon W}{v^{2} k_{0}}\right]^{1 / 2}+\left(1+\frac{\varepsilon W}{2 vk_{0}}\right)\right\}, \end{align} | (69) |
and
Moreover, at
\begin{align} \frac{1}{2} \varepsilon \beta^{2}(0)+\left(v-1\right) \beta(0)+4-\left(v+3\right) \frac{W}{k_{0}}-\frac{1}{2}\frac{ \varepsilon W^{2}}{k_{0}^{2}} = 0, \end{align} | (70) |
\begin{align} \frac{d\lambda}{d z}\bigg|_{z = 0} = 2 k_{0} \sin \varphi(0), \end{align} | (71) |
with
\begin{align} \left(\frac{d \lambda}{d z}\right)^{2}+\frac{W^{2}}{\lambda^{2}}+k_0^{2}v \lambda^{2}+\frac{1}{2} k_0^{2}\varepsilon \lambda^{4} = A. \end{align} | (72) |
Thus, putting (66) with
To test the accuracy of the proposed scheme, with
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
Then, under the same computational environments, but with
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
Furthermore, we simulate the optical bistability by using the proposed finite difference scheme. Firstly, letting the transmittance
Now, we turn to a 2D problem. Setting
\begin{align*} E = \frac{5 \sqrt{2} e^{\mathrm{i} y \sqrt{k_0^{2}+25}}}{\sqrt{\varepsilon} k_0 \cosh (5 x)}. \end{align*} |
In Fig. 7, we exhibit the exact solution and the numerical solution obtained by the new finite difference scheme with
To simulate the transmission and collision of the nonparaxial solitons which are also considered in [3,29], we solve the NLH equation (5) with two different incident waves
\begin{align*} E_{\text {inc }}^1& = \frac{20 \sqrt{2} e^{i y \sqrt{k_{0}^{2}+400}}}{\cosh (20 x)},\\ E_{\text {inc }}^2& = \frac{20 \sqrt{2} e^{i y \sqrt{k_{0}^{2}+400}}}{\cosh (20 x)}+\frac{20 \sqrt{2} e^{i \sqrt{k_{0}^{2}+400}(y / 2-\sqrt{3} x / 2)}}{\cosh [20(x / 2+\sqrt{3} y / 2)]}. \end{align*} |
And the source term is set as
\begin{align*} f = \left\{\begin{array}{c} -\Delta E_{\text {inc }}^l-k_{0}^{2} E_{\text {inc }}^l,(x, y) \in \Omega \backslash \Omega_{0}, \\ 0,\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, y) \in \Omega_{0}, \end{array}\right. l = 1,2. \end{align*} |
The intensities of the incident field and the total field for different cases are shown in Fig. 8 and Fig. 9. When only one incident wave
In this paper, we construct a kind of new finite difference schemes for solving the nonlinear Helmholtz equation based on some iteration methods. Numerical results indicate that, the proposed scheme not only can approximate the high oscillation solution with better computational accuracy, but also can be used to simulate some physical phenomenons in the Kerr medium, such as the optical bistability and the collision of nonparaxial solitons. Moreover, without any extra consideration, this finite difference scheme also provides a route to deal with the problems with discontinuous coefficients or source terms. Thus, it can be extended to much more complex cases, such as the multi-layered Kerr mediums propagating problem and the nonlinear Maxwell's equations.
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1. | Jin Li, Barycentric rational collocation method for semi-infinite domain problems, 2023, 8, 2473-6988, 8756, 10.3934/math.2023439 | |
2. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving KPP equation, 2023, 31, 2688-1594, 3014, 10.3934/era.2023152 | |
3. | Jin Li, Kaiyan Zhao, Xiaoning Su, Selma Gulyaz, Barycentric Interpolation Collocation Method for Solving Fractional Linear Fredholm-Volterra Integro-Differential Equation, 2023, 2023, 2314-8888, 1, 10.1155/2023/7918713 | |
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6. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation, 2023, 31, 2688-1594, 4034, 10.3934/era.2023205 | |
7. | Jin Li, Yongling Cheng, Spectral collocation method for convection-diffusion equation, 2024, 57, 2391-4661, 10.1515/dema-2023-0110 | |
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Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
Algorithm 1 |
Step 1: Give a initial guess Step 2: Solve equation (13) to obtain Step 3: Solve equation (15) to obtain Step 4: Set |
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |