![]() |
64 | 128 | 256 | 512 | |
SAV/CN | Error | 4.75E-3 | 1.17E-3 | 2.91E-4 | 7.17E-5 |
Rate | - | 2.01 | 2.02 | 2.07 | |
SAV/CN + SDC | Error | 1.16E-5 | 6.78E-7 | 4.04E-8 | 2.46E-9 |
Rate | - | 4.07 | 4.03 | 4.02 |
This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the L2 gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.
Citation: Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model[J]. Electronic Research Archive, 2020, 28(2): 1077-1093. doi: 10.3934/era.2020059
[1] | Kai Jiang, Wei Si . High-order energy stable schemes of incommensurate phase-field crystal model. Electronic Research Archive, 2020, 28(2): 1077-1093. doi: 10.3934/era.2020059 |
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This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the L2 gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.
Aperiodic crystals, such as quasicrystals, are an important class of materials whose Fourier spectra cannot be all expressed by a set of basis vectors over the rational number field. The irrational coefficients give rise to the denseness of Fourier spectra which results in the difficulties in the theoretical study. Theoretically, a multiple characteristic length-scale model which possesses, at least, an irrational scale, has been widely applied to study the formation and thermodynamic stability of the aperiodic structures [1, 6, 13, 10, 14, 7]. The early model could trace back to Bak's work on three-dimensional icosahedral quasicrystals. Since then, many related models have been proposed to study aperiodic structures, including for multicomponent systems [7]. Among these models, Lifshitz and Petrich (LP) modified the Swift-Hohenberg model and explicitly added an incommensurate two-length-scale potential into a Lyapunov functional to explore quasiperiodic patterns that emerged in Faraday experiments [13]. Recently, Savitz et al. extended the LP model from two-length-scale potential to multiple (
Recently, various numerical methods have been proposed to solve phase-field equations including the convex splitting methods [17], the linear stabilized schemes [16], the invariant energy quadratization (IEQ) [18] and the scalar auxiliary variable (SAV) approaches [15,11]. The convex splitting method splits the energy functional into the convex and concave parts. The method treats the convex part implicitly and the concave one explicitly to keep the unconditional energy stability. While the application of this method is restricted by the form of the energy functional, such as double-well bulk energy. The linear stabilized scheme adds a penalty term to improve its stability and deals with the nonlinear terms explicitly for implementing it easily. However, such a stabilized approach makes it difficult to design second-order unconditionally energy stable schemes. Assume that the nonlinear part has a lower bound, via introducing an auxiliary variable, the IEQ method transforms the energy functional into a quadratic form to keep the unconditional energy dissipation property. Similarly, the SAV approach introduces a scalar auxiliary variable by supposing the bounded bulk energy and obtains an unconditionally energy stable system. Besides these methods, the spectral deferred correction (SDC) [4,5] algorithm is an efficient strategy to improve the accuracy of the above schemes. In the paper, we will apply the SAV approach to solve the time-dependent equations and further use the SDC strategy to improve the numerical accuracy.
For aperiodic structures, two kinds of numerical methods, including the crystalline approximant method (CAM) and the projection method (PM) are usually used to discretize the quasiperiodic functions [9]. The CAM uses a big periodic structure to approximate an aperiodic structure and corresponds to the Diophantine approximation problem which studies how to approximate irrational numbers by rational numbers [3]. To evaluate aperiodic structures accurately, the CAM needs an extremely big computational region with an unacceptable computational burden to reduce the error of Diophantine approximation. To avoid the Diophantine approximation problem, the PM accurately describes aperiodic structures based on the fact that the aperiodic structure can be regarded as a periodic crystal in an appropriate higher-dimensional space. The PM uses one higher-dimensional periodic region to capture the essential characteristics and greatly reduces computational complexity.
The rest of the paper is organized as follows. In Section 2, we first outline some useful preliminaries of the almost periodic functions and then present the iPFC model. The energy dissipation principle of the
Aperiodic structures are space-filling phases without decay. A useful mathematical theory to describe aperiodic structures is the almost periodic function theory which is a generalization of continuous periodic functions. We define the notation of a
Definition 2.1. Let
|f(r−ζ)−f(r)|<ϵ, for any r∈Rd. |
A function
The almost periodic
⟨f,g⟩AP=limR→∞1|Q(R)|∫Q(R)f(r)¯g(r)dr, |
where
−∫=limR→∞1|Q(R)|∫Q(R). |
Some useful properties of
Proposition 1. (1).
(2). If
These properties can be easily proven from the one-dimensional results [2].
Theorem 2.2. If
⟨f(r),∇g(r)⟩AP=−⟨∇f(r),g(r)⟩AP. |
Proof. Since
bR=limR→∞1|Q(R)|∫∂Q(R)f(r)n⋅¯g(r)ds, |
where
|bR|≤limR→∞M22d(2R)d−1(2R)d=0. |
Therefore, we obtain the desired conclusion by
⟨f(r),∇g(r)⟩AP=−∫f(r)∇g(r)dr=−−∫∇f(r)g(r)dr=−⟨∇f(r),g(r)⟩AP. |
The simplest iPFC model may be the LP model which was originally proposed to study the bi-frequency excited Faraday wave [13]. Concretely, the free energy functional of the LP model can be written as
FLP[ψ(r)]=−∫{c2[(Δ+1)(Δ+q2)ψ]2+(ε2ψ2−α3ψ3+14ψ4)}dr, |
where
F[ψ(r)]=−∫{c2[m∏j=1(Δ+q2j)ψ]2+(ε2ψ2−α3ψ3+14ψ4)}dr, |
where
F[ϕ(r)]=−∫{12[Gϕ]2+N(ϕ)}dr=12‖Gϕ‖2AP+⟨N(ϕ),1⟩AP, |
where
G=m∏j=1(Δ+q2j),N(ϕ)=˜ε2ϕ2−˜α3ϕ3+14ϕ4. |
To solve the iPFC free energy functional, we consider the following Allen-Cahn dynamic equation
ϕt=−W(ϕ),W(ϕ):=δFδϕ=G2ϕ+N′(ϕ). | (1) |
The initial value is
dF(ϕ)dt=⟨δFδϕ,ϕt⟩AP=−‖W(ϕ)‖2AP≤0. |
Then we impose the following mean zero constraint of order parameter on the iPFC model to ensure the mass conservation
−∫ϕ(r)dr=0. | (2) |
In this section, we propose the second-order unconditionally energy stable Crank-Nicolson scheme based on the SAV technique, and also give the error analysis. Further, we use the SDC strategy to improve the temporal accuracy of the second-order scheme.
Let's suppose that
ϕt=−W(ϕ), | (3a) |
W(ϕ)=G2ϕ+R√F1(ϕ)N′(ϕ), | (3b) |
Rt=⟨N′(ϕ)2√F1(ϕ),ϕt⟩AP. | (3c) |
By taking the almost periodic inner products of (3a) with
ddt(12‖Gϕ‖2AP+R2−C1)=−‖W‖2AP≤0. |
The SAV approach can construct high-order unconditionally energy stable schemes. In this section, we discuss a second-order semi-discrete scheme based on the Crank-Nicolson method. Suppose the time interval
Scheme 3.1 (SAV/CN). For
ϕn+1−ϕnτn=−Wn+1/2, | (4a) |
Wn+1/2=G2ϕn+1/2+Rn+1/2√F1(ˉϕn+1/2)N′(ˉϕn+1/2), | (4b) |
Rn+1−Rnτn=⟨N′(ˉϕn+1/2)2√F1(ˉϕn+1/2),ϕn+1−ϕnτn⟩AP, | (4c) |
where
Theorem 3.1. The SAV/CN scheme satisfies the following energy dissipation mechanism
Fn+1SAV/CN−FnSAV/CN≤0, |
where
FnSAV/CN=12‖Gϕn‖2AP+(Rn)2−C1. |
Proof. We take the almost periodic inner products of (4a) with
⟨ϕn+1−ϕn,Wn+1/2⟩AP=−‖Wn+1/2‖2AP, | (5) |
−⟨ϕn+1−ϕn,Wn+1/2⟩AP=−⟨ϕn+1−ϕn,G2ϕn+1/2⟩AP−⟨ϕn+1−ϕn,Rn+1/2√F1(ˉϕn+1/2)N′(ˉϕn+1/2)⟩AP. | (6) |
Multiplying (4c) with
2Rn+1/2(Rn+1−Rn)=⟨Rn+1/2√F1(ˉϕn+1/2)N′(ˉϕn+1/2),ϕn+1−ϕn⟩AP. | (7) |
Adding (5), (6) and (7) together, we obtain
2Rn+1/2(Rn+1−Rn)=−⟨ϕn+1−ϕn,G2ϕn+1/2⟩AP−‖Wn+1/2‖2AP. | (8) |
Since
12(‖Gϕn+1‖2AP−‖Gϕn‖2AP)+(Rn+1)2−(Rn)2=−‖Wn+1/2‖2AP≤0. |
The desired conclusion is obtained from the above equation.
Remark 3.1. It is noted that the modified energy
Remark 3.2(The implementation of the SAV/CN scheme). Denote
u(tn+1/2)=N′(ϕ(tn+1/2))√F1(ϕ(tn+1/2)),un+1/2=N′(ˉϕn+1/2)√F1(ˉϕn+1/2). |
Substituting (4b) and (4c) into (4a), we obtain
ϕn+1−ϕnτn=−[G2ϕn+1/2+un+1/2(Rn+14⟨un+1/2,ϕn+1−ϕn⟩AP)]. | (9) |
Eqn. (9) can be rewritten as
(I+τn2G2)ϕn+1+τn4un+1/2⟨un+1/2,ϕn+1⟩AP=(I−τn2G2)ϕn−τnRnun+1/2+τn4⟨un+1/2,ϕn⟩APun+1/2. | (10) |
Taking the almost periodic inner product with
⟨un+1/2,ϕn+1⟩AP+τn4γn⟨un+1/2,ϕn+1⟩AP=⟨un+1/2,(I+τn2G2)−1cn⟩AP, | (11) |
where
γn=⟨un+1/2,(I+τn2G2)−1un+1/2⟩AP, |
cn=(I−τn2G2)ϕn−τnRnun+1/2+τn4⟨un+1/2,ϕn⟩APun+1/2. |
From (11), we get
⟨un+1/2,ϕn+1⟩AP=⟨un+1/2,(I+12τnG2)−1cn⟩API+14τnγn. | (12) |
Then we can directly calculate
In this section, we will derive the error estimate of the SAV/CN scheme 3.1. Denote
Theorem 3.2. For the Allen-Cahn dynamic equation, assume that
‖Gek‖2AP+(rk)2≤Cτ4∫tk0(‖ϕttt(s)‖2AP+|rttt(s)|2)ds, |
where the constant
Proof. Let's subtract (3) from (4) at
en+1−en=−τwn+1/2+Tn+1/21, | (13) |
wn+1/2=G2en+1/2+Rn+1/2un+1/2−R(tn+1/2)u(tn+1/2), | (14) |
rn+1−rn=12⟨un+1/2,ϕn+1−ϕn⟩AP−12⟨u(tn+1/2),τϕt(tn+1/2)⟩AP+Tn+1/22, | (15) |
where the truncation errors are given by
Tn+1/21=τϕt(tn+1/2)−(ϕ(tn+1)−ϕ(tn)), |
Tn+1/22=τRt(tn+1/2)−(R(tn+1)−R(tn)). |
With the Taylor expansion, the truncation errors can be rewritten as
Tn+1/21=12∫tn+1/2tn+1(tn+1−s)2ϕttt(s)ds−12∫tn+1/2tn(tn−s)2ϕttt(s)ds, |
Tn+1/22=12∫tn+1/2tn+1(tn+1−s)2Rttt(s)ds−12∫tn+1/2tn(tn−s)2Rttt(s)ds. |
Firstly, making the almost periodic inner product of (13) with
⟨en+1−en,wn+1/2⟩AP+τ‖wn+1/2‖2AP=⟨Tn+1/21,wn+1/2⟩AP. | (16) |
Then its right-hand term can be bounded by
⟨Tn+1/21,wn+1/2⟩AP≤τ2‖wn+1/2‖2AP+Cτ‖Tn+1/21‖2AP≤τ2‖wn+1/2‖2AP+Cτ4∫tn+1tn‖ϕttt(s)‖2APds. |
Secondly, by taking the almost periodic inner products of (14) with
−⟨wn+1/2,en+1−en⟩AP=−12(‖Gen+1‖2AP−‖Gen‖2AP)−⟨Rn+1/2un+1/2−R(tn+1/2)u(tn+1/2),en+1−en⟩AP. | (17) |
Without the minus sign, the second term on the right-hand side in the above equation can be transformed into
⟨Rn+1/2un+1/2−R(tn+1/2)u(tn+1/2),en+1−en⟩AP=rn+1/2⟨un+1/2,en+1−en⟩AP+R(tn+1/2)⟨un+1/2−u(tn+1/2),en+1−en⟩AP. | (18) |
Note that
‖un+1/2−u(tn+1/2)‖AP≤C‖N′(ˉϕn+1/2)−N′(ϕ(tn+1/2))‖AP≤C(‖en‖AP+‖en−1‖AP). |
The last term on the right-hand side of (18) can be estimated by
R(tn+1/2)⟨un+1/2−u(tn+1/2),en+1−en⟩AP=R(tn+1/2)⟨un+1/2−u(tn+1/2),−τwn+1/2+Tn+1/21⟩AP≤τ2‖wn+1/2‖2AP+Cτ‖un+1/2−u(tn+1/2)‖2AP+Cτ‖Tn+1/21‖2AP≤τ2‖wn+1/2‖2AP+Cτ(‖en‖2AP+‖en−1‖2AP)+Cτ4∫tn+1tn‖ϕttt(s)‖2APds. |
Thirdly, multiplying the both sides of (15) by
(rn+1)2−(rn)2=rn+1/2⟨un+1/2,ϕn+1−ϕn⟩AP−rn+1/2⟨u(tn+1/2),τϕt(tn+1/2)⟩AP+2rn+1/2Tn+1/22. | (19) |
The first two terms on the right-hand side of (19) can be rewritten as
rn+1/2⟨un+1/2,ϕn+1−ϕn⟩AP−rn+1/2⟨u(tn+1/2),τϕt(tn+1/2)⟩AP=rn+1/2⟨un+1/2,en+1−en⟩AP−rn+1/2⟨u(tn+1/2),Tn+1/21⟩AP+rn+1/2⟨un+1/2−u(tn+1/2),ϕ(tn+1)−ϕ(tn)⟩AP, |
where the last two terms on the right-hand side satisfy
−rn+1/2⟨u(tn+1/2),Tn+1/21⟩AP≤Cτ((rn+1)2+(rn)2)+Cτ4∫tn+1tn‖ϕttt(s)‖2APds, |
and
rn+1/2⟨un+1/2−u(tn+1/2),ϕ(tn+1)−ϕ(tn)⟩AP≤Cτ‖ϕt‖2AP((rn+1/2)2+‖un+1/2−u(tn+1/2)‖2AP)≤Cτ((rn+1)2+(rn)2+‖en‖2AP+‖en−1‖2AP). |
Therefore the last term on the right-hand side of (19) can be bounded by
2rn+1/2Tn+1/22≤Cτ((rn+1)2+(rn)2)+Cτ4∫tn+1tn|rttt(s)|2ds. |
With the above estimates, adding (16), (17) and (19) together leads to
12(‖Gen+1‖2AP−‖Gen‖2AP)+(rn+1)2−(rn)2≤Cτ(‖en‖2AP+‖en−1‖2AP+(rn+1)2+(rn)2)+Cτ4∫tn+1tn‖ϕttt(s)‖2APds+Cτ4∫tn+1tn|rttt(s)|2ds. |
Then we obtain the desired conclusion by summing over
The SDC approach [4,5] is an efficient method to improve the numerical precision for an existing scheme using spectral collocation points. Firstly, we introduce the basic idea of the SDC method. Integrating both sides of the Eqn.(1) with respect to
ϕ(t)=ϕ(0)−∫t0W(ϕ(τ))dτ. | (20) |
Suppose the approximation solution
R[0](t)=ϕ(0)−∫t0W(ϕ[0](τ))dτ−ϕ[0](t), | (21) |
ϵ[0](t)=ϕ(t)−ϕ[0](t). | (22) |
Replacing (22) into (20) yields
ϕ(t)=ϕ(0)−∫t0W(ϕ[0](τ)+ϵ[0](τ))dτ. | (23) |
Then we insert (23) into (22) and subtract (21)
ϵ[0](t)−R[0](t)=−∫t0W(ϕ[0](τ)+ϵ[0](τ))dτ+∫t0W(ϕ[0](τ))dτ. |
By taking the derivative of both sides of the above equation, we obtain
dϵ[0](t)dt=−W(ϕ[0](t)+ϵ[0](t))+W(ϕ[0](t))+dR[0](t)dt. | (24) |
Then, we apply the SDC approach into the second-order SAV/CN scheme to improve the numerical accuracy. We denote this strategy as SAV/CN+SDC. To calculate the integral precisely, we adopt the following Chebyshev nodes,
tn=T2−T2cos(nπNT),n=0,1,⋯,NT. |
The time step size is
ϕn+1[0]−ϕn[0]τn=−Wn+1/2[0],Wn+1/2[0]=G2ϕn+1/2[0]+Rn+1/2[0]√F1(ˉϕn+1/2[0])N′(ˉϕn+1/2[0]),Rn+1[0]−Rn[0]τn=⟨N′(ˉϕn+1/2[0])2√F1(ˉϕn+1/2[0]),ϕn+1[0]−ϕn[0]τn⟩AP, | (25) |
where
ϵn+1[0]−ϵn[0]τn=−Wn+1/2,ϵ[0]+Wn+1/2[0]+Rn+1[0]−Rn[0]τn, | (26a) |
Wn+1/2,ϵ[0]=G2ϕn+1/2,ϵ[0]+Rn+1/2[0]√F1(ˉϕn+1/2[0])N′(ˉϕn+1/2,ϵ[0]), | (26b) |
where
(I+τn2G2)ϵn+1[0]=(I−τn2G2)ϵn[0]−∫tn+1tnW(ϕ[0](τn))dτn−ϕn+1[0]+ϕn[0]−τnRn+1/2[0]√F1(ˉϕn+1/2[0]){N′(ˉϕn+1/2,ϵ[0])−N′(ˉϕn+1/2[0])}. |
A more accurate solution can be updated by
ϕn+1[1]=ϕn+1[0]+ϵn+1[0]. |
The PM is an accurate approach in computing aperiodic structures that can avoid the Diophantine approximation error. The PM is based on the fact that the
ϕ(r)=∑h∈Znˆϕ(h)ei[(P⋅Bh)T⋅r],r∈Rd. |
where
X:={(ˆϕ(h))h∈Zn:ˆϕ(h)∈C,∑h∈Zn|ˆϕ(h)|<∞}. |
In practice, let
XN:={ˆϕ(h)∈X:ˆϕ(h)=0,forall|hj|>Nj/2,j=1,2,…,n}. |
The number of elements in the set is
ˆϕn+1(h)−ˆϕn(h)=−τˆWnh(h),ˆWnh(h)=12m∏j=1[q2j−(PBh)T(PBh)]2[ˆϕn+1(h)+ˆϕn(h)]+Rn+1+Rn2√Fnh1[ˆΦ]^N′nh(h),Rn+1−Rn=∑h1+h2=0^N′nh(h1)2√Fnh1[ˆΦ][ˆϕn+1(h2)−ˆϕn(h2)], |
where
^N′nh(h)=˜εˆϕnh(h)−˜α∑h1+h2=hˆϕnh(h1)ˆϕnh(h2)+∑h1+h2+h3=hˆϕnh(h1)ˆϕnh(h2)ˆϕnh(h3),Fnh1[ˆΦ]=˜ε2∑h1+h2=0ˆϕnh(h1)ˆϕnh(h2)−˜α3∑h1+h2+h3=0ˆϕnh(h1)ˆϕnh(h2)ˆϕnh(h3)+14∑h1+h2+h3+h4=0ˆϕnh(h1)ˆϕnh(h2)ˆϕnh(h3)ˆϕnh(h4)+C1,ˆϕnh(h)=3ˆϕn+1(h)−ˆϕn(h)2. |
In the above equations, the nonlinear terms are
eT1ˆΦ=0, |
where
In this section, we present several numerical examples to verify the accuracy of the SAV/CN and SAV/CN+SDC schemes and to illustrate the advantages of the higher-order scheme in dynamic evolution. We also show the influence of multiple length-scales on the thermodynamic stability of aperiodic structures.
In this subsection, we take the two characteristic length scale iPFC model in one-dimensional space to test the numerical accuracy of the SAV/CN and SAV/CN+SDC schemes. The model parameters are set as
![]() |
64 | 128 | 256 | 512 | |
SAV/CN | Error | 4.75E-3 | 1.17E-3 | 2.91E-4 | 7.17E-5 |
Rate | - | 2.01 | 2.02 | 2.07 | |
SAV/CN + SDC | Error | 1.16E-5 | 6.78E-7 | 4.04E-8 | 2.46E-9 |
Rate | - | 4.07 | 4.03 | 4.02 |
In this subsection, we simulate the dynamic process of
P=(1cos(π/6)cos(π/3)00sin(π/6)sin(π/3)1), |
and the
To show the role of the high-order methods in dynamic evolution, we give the reference energy
In this subsection, we use the dodecagonal quasiperiodic phases as an example to investigate the influence of multi-length-scale potentials on the stability of aperiodic structures. Concretely, we consider three, four, and five multiple length scale iPFC models. The parameters in the potential function are set as
For the iPFC model, we proposed a second-order SAV/CN scheme which is unconditionally energy stable in the almost periodic function sense and gave the error estimate. Meanwhile, we used the SDC approach to further improve the accuracy of the second-order scheme to the fourth-order method through a one-step correction. The PM was applied to discretize spatial functions for computing aperiodic structures to high accuracy. In numerical simulations, the efficiency of the numerical schemes was demonstrated via the numerical convergence rates and the comparison of the dynamic evolutions. By comparing the dynamic evolutions of the DDQCs with different length-scales, we found that increasing the number of characteristic length-scales in the iPFC model significantly is helpful to stabilize aperiodic structures.
This work is supported by National Natural Science Foundation of China (11771368), Hunan Science Foundation of China (2018JJ2376), Youth Project (18B057) and Key Project (19A500) of Education Department of Hunan Province of China, and Hunan Provincial Innovation Foundation for Postgraduate (0943/431000232).
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1. | Duo Cao, Jie Shen, Jie Xu, Computing interface with quasiperiodicity, 2021, 424, 00219991, 109863, 10.1016/j.jcp.2020.109863 | |
2. | Youngjin Hwang, Ildoo Kim, Soobin Kwak, Seokjun Ham, Sangkwon Kim, Junseok Kim, Unconditionally stable monte carlo simulation for solving the multi-dimensional Allen–Cahn equation, 2023, 31, 2688-1594, 5104, 10.3934/era.2023261 |
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64 | 128 | 256 | 512 | |
SAV/CN | Error | 4.75E-3 | 1.17E-3 | 2.91E-4 | 7.17E-5 |
Rate | - | 2.01 | 2.02 | 2.07 | |
SAV/CN + SDC | Error | 1.16E-5 | 6.78E-7 | 4.04E-8 | 2.46E-9 |
Rate | - | 4.07 | 4.03 | 4.02 |