h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.4377e-003 | - | 1.2941e-001 | - |
1/64 | 1.1104e-003 | 4.00 | 6.5173e-002 | 1.99 |
1/128 | 2.7768e-004 | 4.00 | 3.2701e-002 | 1.99 |
1/256 | 6.9427e-005 | 4.00 | 1.6379e-002 | 2.00 |
In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.
Citation: Liping Zhou, Yumei Yan, Ying Liu. Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions[J]. AIMS Mathematics, 2024, 9(10): 27848-27870. doi: 10.3934/math.20241352
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In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.
In recent years, nonclassical boundary and initial-boundary value problems have garnered significant attention across diverse disciplines such as physics, biology, ecology, chemistry, and beyond. Among these, parabolic partial differential equations (PDEs) with nonlocal initial and/or boundary conditions have emerged as powerful tools for modeling a wide array of phenomena. These include, but are not limited to, heat conduction[1], thermoelasticity[2], biotechnology[3], electrochemistry[4], population dynamics [5], and petroleum exploration [6]. The incorporation of nonlocal conditions into these PDEs allows for a more nuanced and realistic representation of the complex interactions and dynamics at play within these systems.
Let QT=Ω×I be the computational domain, where Ω=(0,1)2 and I=(0,T) represent the spatial domain and the time domain, respectively, and T is a positive constant. Here, we consider the following 2D parabolic problem to find a high-accuracy numerical scheme and obtain its theoretical error estimates:
∂u∂t=a2Δu+f(x,y,t),(x,y)∈Ω,t∈(0,T], | (1.1) |
which is subject to the initial conditions
u|t=0=g(x,y),(x,y)∈Ω, | (1.2) |
the Dirichlet boundary conditions
u|x=0=μ1(y,t),y∈(0,1),t∈(0,T], | (1.3) |
u|x=1=μ2(y,t),y∈(0,1),t∈(0,T], | (1.4) |
and the nonlocal boundary conditions
u|y=0=u|y=1+μ3(x,t),x∈(0,1),t∈(0,T], | (1.5) |
uy|y=0=μ4(x,t),x∈(0,1),t∈(0,T], | (1.6) |
where u(x,y,t) is the unknown function, g(x,y), μi(y,t) (i=1,2) and μj(x,t) (j=3,4) are known functions, and a is a positive constant.
These two nonlocal boundary conditions (1.5) and (1.6) are often be used to describe the correlation of a physical quantity across two parallel boundaries in a physical system, as well as the situation where the normal derivative at the boundaries is controlled by external factors, which is commonly used to simulate the interactions between boundaries and boundary effects in processes such as heat conduction and fluid flow.
If the exact solution u of problems (1.1)–(1.6) satisfies certain smootheness conditions, then the compatible condition is deduced as follows: ∀(x,y)∈Ω, the following relations hold:
g(0,y)=μ1(y,0),g(1,y)=μ2(y,0),g(x,0)=g(x,1)+μ3(x,0),gy(x,0)=μ4(x,0). |
The analytical frameworks and numerical techniques employed in tackling parabolic problems with nonlocal conditions have aroused the concern of many scholars. Pertaining to the crucial aspects of convergence and stability for such problems, we acknowledge the foundational work presented in [7,8,9], as well as the extensive references cited therein. Among the prevalent numerical methodologies, finite difference methods (FDM) stand out prominently, with notable contributions from studies such as [7,10,11,12,13]. Additionally, finite element methods (FEM) have garnered substantial attention, exemplified by works cited in [14,15]. Furthermore, the realm of numerical solutions encompasses innovative approaches like Adomian expansions [16], the local coordinates method [17], and the utilization of reproducing kernel spaces [18], each offering unique insights and advancements in this field.
It is widely acknowledged that two-dimensional parabolic partial differential equations (PDEs), characterized by their two spatial variables, pose significant challenges for theoretical analysis, particularly in the realms of convergence analysis and error estimation. The dimensionality of these variables often complicates the mathematical treatment, necessitating innovative strategies. One promising approach to mitigate these difficulties is the utilization of the discrete fourier transform (DFT) method, which offers advantages in reducing the complexity of self-variables during convergence analysis. In this study, we build upon our previous work [19,20] by extending the numerical schemes and integrating the DFT method on the spatial variable x for error estimation within the context of a two-dimensional parabolic PDE subject to a nonlocal boundary condition.
However, a major obstacle arises from the complex boundary condition imposed on the spatial variable y. This condition presents a challenge to traditional DFT methods, which are inherently designed to preserve some boundary conditions. To overcome this limitation, we propose a novel transformation tailored specifically to handle this periodic boundary scenario. Furthermore, we contribute by deriving formulas for the solution derivatives and rigorously proving that these formulas enable us to achieve optimal asymptotic error estimates in the maximum norm. This achievement underscores the effectiveness and applicability of our proposed methodology in accurately approximating and analyzing solutions to two-dimensional parabolic PDEs with intricate boundary conditions.
This paper is organized as follows. In Section 2, the backward Euler difference scheme for the solution of problems (1.1)–(1.6) is presented. Then, in Section 3, we utilize the DFT and develop a new transformation to analyze the error estimate for the corresponding difference equation. The superconvergence for the derivative and its theoretical results are also considered. Finally, some numerical experiments are presented in Section 5.
Now, we use the FDM to discretize problems (1.1)–(1.6). The domain ¯QT is discretized by the uniformly distributed grid points (xi,yj,tn), where
xi=ih,i=0,1,⋯,2N,2Nh=1,yj=jh,j=0,1,⋯,2N,2Nh=1,tn=nτ,n=0,1,⋯,M,Mτ=T, |
where τ is time stepsize, and h is space stepsize along both x and y directions.
Define a function space by
Cm(¯QT)={∂s1+s2+s3u∂xs1∂ys2∂ts3∈C(¯QT)|s1+s2+s3≤m}, |
and its norm by
‖u‖m,∞=maxs1+s2+s3≤m{|∂s1+s2+s3u∂xs1∂ys2∂ts3|},∀(x,y,t)∈¯QT, |
where m and si (i=1,2,3) are given nonnegative integers.
The key to seeking a numerical solution for problems (1.1)–(1.6) lies in how to discretize the nonlocal boundary conditions (1.6). Suppose u∈C4(¯QT), using the Taylor formula, we have
u(x,h,t)=u(x,0,t)+huy(x,0,t)+h22uyy(x,0,t)+h33!uyyy(x,0,t)+O(h4). | (2.1) |
Using (1.1), we have
uyy(x,0,t)=1a2ut(x,0,t)−uxx(x,0)−1a2f(x,0,t). | (2.2) |
Moreover, we obtain
uyyy(x,0,t)=1a2uty(x,0,t)−uxxy(x,0,t)−1a2fy(x,0,t). |
Therefore, with (1.6), we obtain
uyyy(x,0,t)=1a2(μ4)t(x,t)−(μ4)xx(x,t)−1a2fy(x,0,t). | (2.3) |
Substituting (1.6), (2.2), and (2.3) into (2.1), we have
u(x,h,t)=u(x,0,t)+hμ4(x,t)+h22(1a2ut(x,0,t)−uxx(x,0,t)−1a2f(x,0))+h33!(1a2(μ4)t(x,t)−(μ4)xx(x,t)−1a2fy(x,0,t))+O(h4), |
i.e.
ut(x,0,t)=2a2h2(u(x,h,t)−u(x,0,t))+a2uxx(x,0,t)+˜μ4(x,t)+O(h2), | (2.4) |
where
˜μ4(x,t)=f(x,0,t)−2a2hμ4(x,t)−h3((μ4)t(x,t)−a2(μ4)xx(x,t)−fy(x,0,t)). | (2.5) |
From the derivation process described above, the discretization of (1.6) is converted to discretizing (2.4).
Let uni,j and Uni,j be the exact value and the approximation of u(x,y,t) at grid point (xi,yj,tn), respectively. Let fni,j=f(xi,yj,tn), gi,j=g(xi,yj), (μm)nj=μm(yj,tn) (m=1,2), (μ3)ni=μ3(xi,tn) and (˜μ4)ni=˜μ4(xi,tn).
Then, (2.4) is approximated by the following difference equations:
Uni,0−Un−1i,0τ=2a2h2(Uni,1−Uni,0)+a2Uni+1,0−2Uni,0+Uni−1,0h2+2(μ4)nih+(˜μ4)ni,i=1,2,⋯,2N−1. | (2.6) |
Also, we obtain the difference equations of (1.1)
Uni,j−Un−1i,jτ=a2(Uni−1,j−2Uni,j+Uni+1,jh2+Uni,j−1−2Uni,j+Uni,j+1h2)+fni,j,i,j=1,2,⋯,2N−1,n=1,2,⋯,M. | (2.7) |
Let αni,0 be the local truncature error of (2.6). When u∈C4(¯QT), using the Taylor formula, we can easily deduce that
|αni,0|≲‖u‖4,∞(τ+h2)≲τ+h2,i=1,2,⋯,2N−1. |
Similarly, when u∈C4(¯QT), it holds that
|αni,j|≲‖u‖4,∞(τ+h2)≲τ+h2,i=1,2,⋯,2N−1,j=1,2,⋯,2N−1, |
where αni,j is the local truncation error of (2.7).
Moreover, we obtain
|αni,j|≲τ+h2,i=1,2,⋯,2N−1,j=0,1,⋯,2N−1. | (2.8) |
From the above, we obtain the backward Euler difference scheme of problems (1.1)–(1.6).
Uni,j−Un−1i,jτ=a2(Uni−1,j−2Uni,j+Uni+1,jh2+Uni,j−1−2Uni,j+Uni,j+1h2)+fni,j,i,j=1,2,⋯,2N−1,n=1,2,⋯,M, | (2.9a) |
U0i,j=gi,j,i,j=0,1,⋯,2N, | (2.9b) |
Un0,j=(μ1)nj,j=0,1,⋯,2N,n=1,2,⋯,M, | (2.9c) |
Un2N,j=(μ2)nj,j=0,1,⋯,2N,n=1,2,⋯,M, | (2.9d) |
Uni,0=Uni,2N+(μ3)ni,i=1,2,⋯,2N−1,n=1,2,⋯,M, | (2.9e) |
Uni,0−Un−1i,0τ=2a2h2(Uni,1−Uni,0)+a2h2(Uni−1,0−2Uni,0+Uni+1,0)+(˜μ4)ni,i=1,2,⋯,2N−1,n=1,2,⋯,M. | (2.9f) |
Let eni,j=uni,j−Uni,j be the error of the approximation solution U at the grid point (xi,yj,tn), and μ=τh2 be the grid ratio. Then, the error equations of (2.9a)–(2.9f) are
eni,j−en−1i,j=a2μ(eni−1,j+eni+1,j+eni,j−1+eni,j+1−4eni,j)+ταni,j,i,j=1,⋯,2N−1,n=1,2,⋯,M, | (3.1a) |
e0i,j=0,i,j=0,1,⋯,2N, | (3.1b) |
en0,j=0,j=0,1,⋯,2N,n=1,2,⋯,M, | (3.1c) |
en2N,j=0,j=0,1,⋯,2N,n=1,2,⋯,M, | (3.1d) |
eni,0=eni,2N,i=1,2,⋯,2N−1,n=1,2,⋯,M, | (3.1e) |
eni,0−en−1i,0=2a2μ(eni,1−eni,0)+a2μ(eni−1,0−2eni,0+eni+1,0)+ταni,0,i=1,2,⋯,2N−1,n=1,2,⋯,M. | (3.1f) |
Given the complexity of the above error equations, the key to obtaining an error estimate lies in finding transformations that separate the index variables i, j, and n.
Since the error sequence {eni,j} satisfies (3.1c) and (3.1d), applying the DFT to {eni,j} with respect to i, we obtain
eni,j=√2h2N−1∑k=1ˆenk,jsin(kπxi),i,j=0,1,⋯,2N. | (3.2) |
Similarly, applying the DFT to {αni,j} with respect to i, we obtain
αni,j=√2h2N−1∑k=1ˆαnk,jsin(kπxi),i=1,2,⋯,2N−1,j=0,1,⋯,2N−1. | (3.3) |
It follows from (2.8) and (3.3) that
|ˆαnk,j|≲τ+h2h12,k=1,2,⋯,2N−1,j=0,1,⋯,2N−1. | (3.4) |
Substituting (3.2) and (3.3) into (3.1a), we obtain
√2h2N−1∑k=1(ˆenk,j−ˆen−1k,j)sin(kπxi)=√2ha2μ2N−1∑k=1(ˆenk,j(sin(kπxi−1)−2sin(kπxi)+sin(kπxi+1))+(ˆenk,j−1−2ˆenk,j+ˆenk,j+1)sin(kπxi))+√2hτ2N−1∑k=1ˆαnk,jsin(kπxi),i,j=1,2,⋯,2N−1. | (3.5) |
Utilizing the properties of the DFT and (3.5), we obtain
ˆenk,j−ˆen−1k,j=a2μ(ˆenk,j−1−(2+4sin2kπh2)ˆenk,j+ˆenk,j+1)+τˆαnk,j,j=1,2,⋯,2N−1. | (3.6) |
Similarly, substituting (3.2) and (3.3) into (3.1f), we have
ˆenk,0−ˆen−1k,0=2a2μ(ˆenk,1−ˆenk,0)−4a2μsin2kπh2ˆenk,0+τˆαnk,0. | (3.7) |
Substituting (3.2) into (3.1b) and (3.1e), we deduce that
ˆe0k,j=0,j=0,1,⋯,2N, | (3.8) |
and
ˆenk,0=ˆenk,2N. | (3.9) |
Given that the sequence {ˆenk,j} adheres to the condition specified in (3.9), the conventional DFT is found to be inadequate for our analytical needs. In pursuit of a suitable tool for analysis, we aspire for a novel transformation that not only fulfills the criteria outlined in (3.8) but also possesses the property of invertibility. Drawing inspiration from the formulation of the DFT, we introduce a fresh transformation tailored specifically for the sequence {ˆenk,j} with respect to j in the following way, aiming to address the aforementioned limitations and meet our analytical needs.
ˆenk,j=2N−1∑l=0˜enk,lTl(yj),j=0,1,⋯,2N−1, | (3.10) |
where
Tl(y)={cos(2lπy),l=0,1,⋯,N,ysin(2lπy),l=N+1,N+2,⋯,2N−1. | (3.11) |
It is straightforward to verify Lemma 3.1.
Lemma 3.1. The sequence {Tl(yj)} has the following properties.
(1) Tl(y0)={1,l=0,1,⋯,N,0,l=N+1,N+2,⋯,2N−1.
(2) Tl(y1)−Tl(y0)={(cos(2lπh)−1)Tl(y0),l=0,1,⋯,N,hsin(2lπh),l=N+1,N+2,⋯,2N−1.
(3) For any 0≤l,j≤2N−1,
Tl(yj−1)−2Tl(yj)+Tl(yj+1)={2(cos(2lπh)−1)Tl(yj),l=0,1,⋯,N,2(cos(2lπh)−1)Tl(yj)+2hT2N−l(yj)sin(2lπh),l=N+1,N+2,⋯,2N−1. |
(4) Tl(y2N−j)={cos(2lπyj),l=0,1,⋯,N,(yj−1)sin(2lπyj),l=N+1,N+2,⋯,2N−1.
For the sake of simplicity in the subsequent analysis, we introduce
Pl(y):={cos(2lπy),l=0,1,⋯,N,sin(2lπy),l=N+1,N+2,⋯,2N−1, | (3.12) |
and consider the orthogonality relation of the polynomials Pl(yi) and Pl(yj).
Lemma 3.2. Given that i,j=0,⋯,N, we have the following identity:
N∑l=0σlPl(yi)Pl(yj)={N2σi,i=j,0,i≠j, | (3.13) |
where
σl={12,l=0,N,1,otherwise. | (3.14) |
Proof. Using (3.14) and (3.12), and noticing 2Nh=1 and yi=ih, we have
N∑l=0σlPl(yi)Pl(yj)=N−1∑l=1Pl(yi)Pl(yj)+12∑l=0,NPl(yi)Pl(yj)=N−1∑l=1cos(2lπyi)cos(2lπyj)+12(1+cos(2Nπyi)cos(2Nπyj))=12N−1∑l=1cos(2lπyi+j)+12N−1∑l=1cos(2lπyi−j)+1+(−1)i+j2. | (3.15) |
For 0≤m≤2N, we have
2N−1∑l=1cos(2lπym)=N−1∑l=1(cos(2(l−1)πym)+cos(2(l+1)πym))−1+cos(2(N−1)πym)+cos(2πym)−cos(2Nπym)=2cos(2πym)N−1∑l=1cos(2lπym)+(1+(−1)m)(cos(2πym)−1), |
i.e.,
2(1−cos(2lπym))N−1∑l=1cos(2lπym)=(1+(−1)m)(cos(2πym)−1). |
If cos(2lπym)≠1, then
N−1∑l=1cos(2lπym)=−1+(−1)m2. | (3.16) |
Now we focus on the case i≠j. Since 0≤i,j≤N, it follows that 0<yi+j<1 and 0<|yi−j|<1. Furthermore, when l ranges from 1 to N−1, we observe that cos(2lπyi−j)≠1 and cos(2lπyi+j)≠1.
Thus, with (3.16), and observing the same parity of i+j and i−j, we obtain
N−1∑l=1cos(2lπyi+j)=N−1∑l=1cos(2lπyi−j)=−1+(−1)i+j2. |
Substituting the above equality into (3.15), we deduce that
N∑l=0σlPl(yi)Pl(yj)=0,i≠j. | (3.17) |
In the next, we consider the case i=j. Noting that when 1≤l≤N−1, cos(2lπy2i) is equal to 1 only in i=0 and i=N. Therefore, by utilizing (3.16), we obtain
N−1∑l=1cos(2lπy2i)={−1,i=1,2,⋯,N−1,N−1,i=0,N. | (3.18) |
Substituting (3.18) into (3.15), we deduce that
N∑l=0σlPl(yi)Pl(yj)={N2,i=1,2,⋯,N−1,N,i=0,N. | (3.19) |
Furthermore, with (3.14), (3.17), and (3.19), we arrive at the conclusion stated in (3.13).
Similar to Lemma 3.2, we can derive the subsequent lemma as well.
Lemma 3.3. Given that i,j=N+1,⋯,2N−1, we have the following identity
2N−1∑l=N+1Pl(yi)Pl(yj)={N2,i=j,0,i≠j. | (3.20) |
Therefore, we can conclude the following lemma.
Lemma 3.4. Suppose
ai=2N−1∑l=N+1ˆalPl(yi),i=N+1,N+2,⋯,2N−1. | (3.21) |
Then
ˆal=2N2N−1∑i=N+1aiPl(yi),l=N+1,N+2,⋯,2N−1. | (3.22) |
Proof. Using (3.21), (3.12), and Lemma 3.3, we obtain
2N−1∑i=N+1aiPl(yi)=2N−1∑i=N+12N−1∑m=N+1ˆamPl(yi)Pm(yi)=2N−1∑m=N+1ˆam2N−1∑i=N+1Pi(yl)Pi(ym)=N2ˆal. |
The proof is finished.
Similar to Lemma 3.4, we obtain
Lemma 3.5. Suppose
ai=N∑l=0ˆalPl(yi),i=0,1,⋯,N. | (3.23) |
Then
ˆal=2σlNN∑i=0σiaiPl(yi),l=0,1,⋯,N. | (3.24) |
Based on Lemmas 3.4 and 3.5, we obtain the invertible transformation of (3.25).
Lemma 3.6. Suppose
ˆaj=2N−1∑l=0˜alTl(yj),j=0,1,⋯,2N−1. | (3.25) |
Then,
˜al={2σlNN∑j=0σj((1−yj)ˆaj+yjˆan2N−j)cos(2lπyj),l=0,1,⋯,N,2NN∑j=0(ˆaj−ˆa2N−j)sin(2lπyj),l=N+1,N+2,⋯,2N−1, | (3.26) |
where
σj={12,j=0,N,1,otherwise. | (3.27) |
Proof. Using (3.25), (3.12), and Lemma 3.1, we have
ˆa2N−j=N∑l=0˜alPl(yj)+(yj−1)2N−1∑l=N+1˜alPl(yj), |
and
ˆaj=N∑l=0˜alPl(yj)+yj2N−1∑l=N+1˜alPl(yj). |
From the two equalities above, it follows that
(1−yj)ˆaj+yjˆa2N−j=N∑l=0˜alPl(yj) |
and
ˆaj−ˆa2N−j=2N−1∑l=N+1˜alPl(yj). |
Moreover, using Lemmas 3.5 and 3.4, we arrive at the conclusion stated in (3.26).
Similar to (3.10), we use the same transformation to {ˆαnk,j} with respect to j in the following way:
ˆαnk,j=2N−1∑l=0˜αnk,lTl(yj),j=0,1,⋯,2N−1. | (3.28) |
Using (3.4), (3.27), and Lemma 3.6, and noting that 0≤yj≤1 (j=0,1,⋯,2N), we can deduce
|˜αnk,l|≲τ+h2h12,l=0,1,⋯,2N−1. | (3.29) |
Substituting (3.10) and (3.28) into (3.6), we obtain
2N−1∑l=0(˜enk,l−˜en−1k,l)Tl(yj)=a2μ2N−1∑l=0˜enk,l(Tl(yj−1)−(2+4sin2kπh2)Tl(yj)+Tl(yj+1))+τ2N−1∑l=0˜αnk,lTl(yj),j=1,2,⋯,2N−1. | (3.30) |
Using Lemma 1, (3.30) can be rewritten as
2N−1∑l=0(˜enk,l−˜en−1k,l)Tl(yj)=a2μ(2N−1∑l=0(2(cos(2lπh)−1)−4sin2kπh2)˜enk,lTl(yj)+2h2N−1∑l=N+1˜enk,lT2N−l(yj)sin(2lπh))+τ2N−1∑l=0˜αnk,lTl(yj),j=1,2,⋯,2N−1. |
Let l:=2N−l in 2N−1∑l=N+1˜enk,lT2N−l(yj)sin(2lπh), the above equalities have the following form:
2N−1∑l=0(˜enk,l−˜en−1k,l)Tl(yj)=−2a2μ(22N−1∑l=0(sin2(lπh)+sin2kπh2)˜enk,lTl(yj)+hN−1∑l=1˜enk,2N−lTl(yj)sin(2lπh))+τ2N−1∑l=0˜αnk,lTl(yj),j=1,2,⋯,2N−1. | (3.31) |
Substitute (3.10) into (3.7), then
2N−1∑l=0(˜enk,0−˜en−1k,0)Tl(y0)=2a2μ2N−1∑l=0˜enk,l(Tl(y1)−Tl(y0))−4a2μsin2kπh22N−1∑l=0˜enk,lTl(y0)+τ2N−1∑l=0˜αnk,lTl(y0). |
Moreover, using Lemma 3.1, we obtain
2N−1∑l=0(˜enk,0−˜en−1k,0)Tl(y0)=−2a2μ(22N−1∑l=0(sin2(lπh)+sin2kπh2)˜enk,lTl(y0)+hN−1∑l=1˜enk,2N−lTl(y0)sin(2lπh))+τ2N−1∑l=0˜αnk,lTl(y0). |
Through comparing with the above equality and (3.31), we find that (3.31) also holds for j=0. Therefore,
2N−1∑l=0(˜enk,l−˜en−1k,l)Tl(yj)=−2a2μ(22N−1∑l=0(sin2(lπh)+sin2kπh2)˜enk,lTl(yj)+hN−1∑l=1˜enk,2N−lTl(yj)sin(2lπh))+τ2N−1∑l=0˜αnk,lTl(yj),j=0,1,⋯,2N−1. | (3.32) |
Using Lemma 3.6 to perform an invertible transformation on (3.32), we obtain
˜enk,l−˜en−1k,l={−4a2μ(sin2(lπh)+sin2kπh2)˜enk,l−2a2μh˜enk,2N−lsin(2lπh)+τ˜αnk,l,l=1,2,⋯,N−1,−4a2μ(sin2(lπh)+sin2kπh2)˜enk,l+τ˜αnk,l,l=0,N,N+1,⋯,2N−1. | (3.33) |
Let
ωk,l=11+4a2μ(sin2(lπh)+sin2kπh2). | (3.34) |
Obviously,
0<ωk,l<1. | (3.35) |
Using (3.34), (3.33) can be rewritten as
˜enk,l={ωk,l˜en−1k,l−2a2μhωk,l˜enk,2N−lsin(2lπh)+τωk,l˜αnk,l,l=1,2,⋯,N−1,ωk,l˜en−1k,l+τωk,l˜αnk,l,l=0,N,N+1,⋯,2N−1. | (3.36) |
Substituting (3.10) into (3.8), and using Lemma 3.6, we can easily deduce that
˜enk,l=0,l=0,1,⋯,2N−1. | (3.37) |
Using (3.36) and (3.37), we obtain the following recursive formula for {˜enk,l}:
˜enk,l={−2a2μhsin(2lπh)n∑m=1(ωk,l)n−m+1˜emk,2N−l+τn∑m=1(ωk,l)n−m+1˜αmk,l,l=1,2,⋯,N−1,τn∑m=1(ωk,l)n−m+1˜αmk,l,l=0,N,N+1,⋯,2N−1. | (3.38) |
In order to estimate {˜enk,l}, we first prove the following estimation
n∑m=1(ωk,l)n−m+1≲{1τ(l2+k2),l=0,1,⋯,N,1τ((2N−l)2+k2),l=N+1,N+2,⋯,2N−1. | (3.39) |
In fact, from (3.34) and (3.35), we can derive that
n∑m=1(ωk,l)n−m+1=ωk,l−(ωk,l)n+11−ωk,l=1−(ωk,l)nμ(sin2(lπh)+sin2kπh2)≤1μ(sin2(lπh)+sin2kπh2). | (3.40) |
For 0≤l≤N, we have lπh∈[0,π2]. Observe that kπh2∈(0,π2)(1≤k≤2N−1). Therefore, (3.40) can be rewritten as
n∑m=1(ωk,l)n−m+1≲1μh2(4l2+k2)≲1τ(l2+k2). | (3.41) |
For N≤l≤2N−1, from 2Nh=1 and 0<(2N−l)πh≤π2, we have
sin(lπh)=sin(2(2N−l)πh)≥2(2N−l)h. |
Moreover, (3.40) can be written as
n∑m=1(ωk,l)n−m+1≲1μh2(4(2N−l)2+k2)≲1τ((2N−l)2+k2). | (3.42) |
Therefore, combining (3.41) with (3.42), (3.39) holds.
Now, we give the estimation of {˜enk,l} in three cases.
Case 1. N≤l≤2N−1
With (3.38), (3.35), (3.29), and (3.39), we have
|˜enk,l|≤τn∑m=1(ωk,l)n−m+1|˜αnk,l|≲τ(τ+h2)h12n∑m=1(ωk,l)n−m+1≲τ+h2h12((2N−l)2+k2). | (3.43) |
Case 2. 1≤l≤N−1
Using (3.43), we obtain
|˜enk,2N−l|≲τ+h2h12(l2+k2). | (3.44) |
From the above inequality, and using (3.38), (3.35), (3.39), (3.29), and μ=τh2, we obtain
|˜enk,l|≤2a2μhsin(2lπh)n∑m=1(ωk,l)n−m+1|˜emk,2N−l|+τn∑m=1(ωk,l)n−m+1|˜αmk,l|≲(μhsin(2lπh)⋅τ+h2h12(l2+k2)+τ(τ+h2)h12)n∑m=1(ωk,l)n−m+1≲τ+h2h12(l2+k2)(sin(lπh)h(l2+k2)+1)≲τ+h2h12(l2+k2). | (3.45) |
Case 3. l=0
Observing that ωk,0=11+4a2μsin2kπh2, similar to deduce (3.43), we obtain
|˜enk,0|≲τ+h2k2h12. | (3.45 = 6) |
From (3.10), (3.43), (3.45), and (3.46), and noticing that Tl(yj) is bounded for any 0≤l,j≤2N−1, we have
|ˆenk,j|≤2N−1∑l=0|˜enk,l||Tl(yj)|≲τ+h2k2h12(1k2+N−1∑l=11l2+k2+2N−1∑l=N1(2N−l)2+k2)≲τ+h2h12N−1∑l=01l2+k2. | (3.47) |
Furthermore, from (3.2), we obtain
|eni,j|≤√2h2N−1∑k=1|ˆenk,j|≲(τ+h2)2N−1∑k=1N−1∑l=01l2+k2≲(τ+h2)(32N∑k=11k2+2N∑k=22N∑l=21l2+k2)≲(τ+h2)2N∑k=22N∑l=21l2+k2. | (3.48) |
Since 1x2+y2 is increasing monotonically with respect to variables x and y for x,y>0, respectively, it follows that
2N∑k=22N∑l=21l2+k2=h22N∑k=22N∑l=21(lh)2+(kh)2≤∬Ωh1x2+y2dxdy<∫π20dθ∫√2hdrr≤π2|lnh|, | (3.49) |
where Ωh=[h,1]×[h,1].
Using (3.49) and (3.48), and noticing (3.1c)–(3.1e), we can obtain the following error estimation theorem.
Theorem 3.1. Suppose u∈C4(¯QT). For any postive integer 1≤n≤M, the following estimates for (2.9a)–(2.9f)
|eni,j|≲(τ+h2)|lnh|,i,j=0,1,⋯,2N. |
hold.
In this section, we present the approximation formulas for the partial derivatives of u with respect to two spatial variables, which exhibit superconvergence under certain smooth conditions.
Let Ux and Uy be the approximation functions for the partial derivatives ux and uy, respectively. For any tn (1≤n≤M), we introduce the following approximation formulas for ux and uy at the grid point (xi,yj,tn), respectively:
Ux(xi,yj,tn)=Uni+1,j−Uni−1,j2h,1≤i≤2N−1,0≤j≤2N, | (4.1) |
and
Uy(xi,yj,tn)=Uni,j+1−Uni,j−12h,0≤i≤2N,1≤j≤2N−1. | (4.2) |
Before exploring the superconvergence of (4.1) and (4.2), we first present the following lemma.
Lemma 4.1. Suppose that the function p(x)∈C1[0,1] satisfies
maxx∈[0,1]{|p(x)|,|p′(x)|}≤M, | (4.3) |
where M is a positive constant. If
ˆpk=√2h2N−1∑i=1pisin(iπxk),i=1,2,⋯,2N−1, | (4.4) |
then
|ˆpk|≤Mπkh12≲1kh12. | (4.5) |
Proof. Let θk=kπh2. Obviously, θk∈(0,π2). We can easily verify the following equality:
2N−1∑i=1(pi−1−2pi+pi+1)sin(iπxk)=2N−1∑i=1pi(sin((i+1)πxk)−2sin(iπxk)+sin((i−1)πxk))+p0sin(πxk)+p2Nsin((2N−1)πxk)=−4sin2θk2N−1∑i=1pisin(iπxk)+(p0+(−1)kp2N)sin(2θk). | (4.6) |
Noting that
2N−1∑i=1(pi−pi−1)sin(iπxk)=2N−1∑i=1(∫xixi−1p′(x)sin(kπx)dx+∫xixi−1p′(x)(sin(kπxi)−sin(kπx))dx)=∫x2N−10p′(x)sinkπxdx+22N−1∑i=1∫xixi−1p′(x)coskπ(xi+x)2sinkπ(xi−x)2dx, | (4.7) |
the following equality is also verified:
2N−1∑i=1(pi+1−pi)sin(iπxk)=∫1x1p′(x)sin(kπx)dx+22N−1∑i=1∫xi+1xip′(x)coskπ(xi+1+x)2sinkπ(xi+1−x)2dx. | (4.8) |
Subtracting (4.7) from (4.8), we have
2N−1∑i=1(pi−1−2pi+pi+1)sin(iπxk)=∫1x2N−1p′(x)sin(kπx)dx−∫x10p′(x)sin(kπx)dx+2∫1x2N−1p′(x)coskπ(1+x)2sinkπ(1−x)2dx−2∫x10p′(x)coskπ(x1+x)2sinkπ(x1−x)2dx. |
Thus, using (4.3), we obtain
|2N−1∑i=1(pi−1−2pi+pi+1)sin(iπxk)|≤6Mh. | (4.9) |
Using (4.6), (4.3), and (4.9), and noting θk∈(0,π2), we obtain
|2N−1∑i=1pisin(iπxk)|=14sin2θk|2N−1∑i=1(pi−1−2pi+pi+1)sin(iπxk)−(p0+(−1)kp2N)sin(2θk)|≤3Mh+2Msinθk2sin2θk≤Mπkh. |
Therefore, the lemma is proved with (4.4).
Next, we study the superconvergence of (4.1).
Theorem 4.1. Suppose u∈C5(¯QT). Then, for any integer 1≤n≤M,
|Ux(xi,yj,tn)−ux(xi,yj,tn)|≲(τ+h2)|lnh|,i=1,2,⋯,2N−1,j=0,1,⋯,2N. | (4.10) |
Proof. From (4.1) and u∈C5(¯QT), we obtain
Ux(xi,yj,tn)=ux(xi,yj,tn)+eni+1,j−eni−1,j2h+O(h2). |
Thus, in order to prove this theorem, it suffices to prove the following inequality, i.e., for any given integer 1≤n≤M,
|eni+1,j−eni−1,j2h|≲(τ+h2)|lnh|,i=1,2,⋯,2N−1,j=0,1,⋯,2N. | (4.11) |
From (3.2) and (3.10), we have
eni+1,j−eni−1,j2h=2√2h2N−1∑k=1ˆenk,jsin(kπh)cos(kπxi)=2√2h2N−1∑k=12N−1∑l=0˜enk,lTl(yj)sin(kπh)cos(kπxi). | (4.12) |
Since u∈C5(¯QT), using Lemma 4.1, (2.8), and (3.3), we obtain
|ˆαnk,j|≲τ+h2kh12. | (4.13) |
Correspondingly, (3.29) is written as
|˜αnk,l|≲τ+h2kh12. | (4.14) |
For this, by modifying (3.43), (3.45), and (3.46), we obtain
|˜enk,l|≲{τ+h2kh12(l2+k2),0≤l≤N,τ+h2kh12((2N−l)2+k2),N+1≤l≤2N−1. | (4.15) |
Using (4.12) and (4.15), and given |Tl(y)|≤1 for y∈[0,1], we obtain
|eni+1,j−eni−1,j2h|≲h122N−1∑k=1k2N−1∑l=0|˜enk,l|≤(τ+h2)2N−1∑k=1(N−1∑l=01l2+k2+2N−1∑l=N1(2N−l)2+k2)≲(τ+h2)2N∑k=22N∑l=21l2+k2. | (4.16) |
From this, using (3.49), we prove (4.11). Therefore, (4.10) holds.
In the following, we discuss the superconvergence properties of (4.2).
Theorem 4.2. Suppose u∈C5(¯QT). Then, for any integer 1≤n≤M,
|Uy(xi,yj,tn)−uy(xi,yj,tn)|≲(τ+h2)ln2h,i=0,1,⋯,2N,j=1,2,⋯,2N−1, | (4.17) |
hold.
Proof. From (4.2) and u∈C5(¯QT), we obtain
Ux(xi,yj,tn)=ux(xi,yj,tn)+eni,j+1−eni−1,j−12h+O(h2). | (4.18) |
From (4.18), in order to prove this theorem, we only need to prove that for any integer 1≤n≤M,
|eni,j+1−eni,j−12h|≲(τ+h2)ln2h,i=0,1,⋯,2N,j=1,2,⋯,2N−1. | (4.19) |
Using (3.2) and (3.10), and noting that T0(yj+1)−T0(yj−1)=0, we find that
eni,j+1−eni,j−12h=2√2h2N−1∑k=1(ˆenk,j+1−ˆenk,j−1)sin(kπxi)=2√2h2N−1∑k=12N−1∑l=0˜enk,l(Tl(yj+1)−Tl(yj−1))sin(kπxi)=2√2h2N−1∑k=12N−1∑l=1˜enk,l(Tl(yj+1)−Tl(yj−1))sin(kπxi). | (4.20) |
Using (3.11), when N≤l≤2N−1, we have
|Tl(yj+1)−Tl(yj)|≤2|yjcos((yj+1+yj)lπ)sin(2lπh)|+h|sin(2lπyj+1)|≲|sin((2N−l)πh)|+h≲(2N−l)h. | (4.21) |
Furthermore, we also deduce that
|Tl(yj+1)−Tl(yj)|=2|sin((yj+1+yj)lπ)sin(2lπh)|≲lh. | (4.22) |
Using (4.20), (4.15), (4.21), and (4.22), we obtain
|eni,j+1−eni,j−12h|≲1h122N−1∑k=12N−1∑l=1|˜enk,l||Tl(yj+1)−Tl(yj−1)|≲τ+h2h2N−1∑k=1(N∑l=1lhk(k2+l2)+2N−1∑l=N+1(2N−l)hk(k2+(2N−l)2))≲(τ+h2)2N−1∑k=1N∑l=1lk(k2+l2)≲(τ+h2)(2N∑l=1l1+l2+2N∑k=11k(k2+1)+2N∑k=22N∑l=2lk(k2+l2))≲(τ+h2)(|lnh|+2N∑k=22N∑l=2lk(k2+l2)). | (4.23) |
Upon observing
22N∑k=22N∑l=2lk(k2+l2)=2N∑k=22N∑l=2(lk(k2+l2)+kl(k2+l2))=2N∑k=22N∑l=21kl≤ln2h, |
by substituting this result into (4.23), we obtain
|eni,j+1−eni,j−12h|≲(τ+h2)ln2h. |
This result confirms (4.19), thereby establishing the validity of (4.17).
In this section, we present two numerical examples to validate the theoretical results and investigate the efficiency and the superconvergence properties of the numerical schemes. Our aim is to demonstrate the practical implications of the theoretical findings and assess the performance of the proposed methods. Let
‖U−u‖∞:=max1≤n≤M0≤i,j≤2N|Uni,j−uni,j|,‖Ux−ux‖∞:=max1≤n≤M1≤i≤2N−10≤j≤2N|(Ux)ni,j−(ux)ni,j|,‖Uy−uy‖∞:=max1≤n≤M0≤i≤2N1≤j≤2N−1|(Uy)ni,j−(uy)ni,j|. |
Example 5.1. In (1.1)–(1.6), take
a=1,T=1,f(x,y,t)=0,g(x,y)=ex+y,μ1(y,t)=ey+2t,μ2(y,t)=e1+y+2t,μ3(x,t)=ex+2t(1−e),μ4(x,t)=ex+2t. |
The exact solution is u=ex+y+2t which can be easily verified.
The results are reported in Tables 1–3. From Table 1, we can observe that in the cases of τ=h2 and τ=h, the error ‖U−u‖∞ is approximately of the order O(h2) and O(h), respectively. This observation verifies the correctness of Theorem 3.1.
h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.4377e-003 | - | 1.2941e-001 | - |
1/64 | 1.1104e-003 | 4.00 | 6.5173e-002 | 1.99 |
1/128 | 2.7768e-004 | 4.00 | 3.2701e-002 | 1.99 |
1/256 | 6.9427e-005 | 4.00 | 1.6379e-002 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 1.7481e-002 | - | 7.0681e-003 | - |
1/64 | 1/642 | 4.4901e-003 | 3.89 | 1.8960e-003 | 3.73 |
1/128 | 1/1282 | 1.1379e-003 | 3.95 | 5.0110e-004 | 3.78 |
1/256 | 1/2562 | 2.8640e-004 | 3.97 | 1.3025e-004 | 3.85 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 6.1798e-001 | - | 1.7722e-001 | - |
1/64 | 1/64 | 3.3247e-001 | 1.86 | 1.0037e-001 | 1.77 |
1/128 | 1/128 | 1.7242e-001 | 1.93 | 5.3329e-002 | 1.88 |
1/256 | 1/256 | 8.7795e-002 | 1.96 | 2.7478e-002 | 1.94 |
Furthermore, from Tables 2 and 3, it is evident that when τ=h2, both ‖Ux−ux‖∞ and ‖Uy−uy‖∞ are close to the order O(h2). On the other hand, when τ=h, ‖Ux−ux‖∞ and ‖Uy−uy‖∞ approach the order O(h). These findings support the theoretical expectations regarding the convergence rates of the spatial derivatives. Therefore, the correctness of Theorems 4.1 and 4.2 is verified.
Example 5.2. In problems (1.1)–(1.6), take
a=1,T=1,f(x,y,t)=0,g(x,y)=(1+y)ex,μ1(y,t)=(1+y)et,μ2(y,t)=(1+y)e1+t,μ3(x,t)=−ex+t,μ4(x,t)=ex+t. |
It is easily verified that its exact solution is u=(1+y)ex+t.
Numerical results for Example 5.2 are reported in Tables 4–6. These results verify the correctness of Theorems 3.1, 4.1 and 4.2 again.
h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.2726e-004 | - | 1.1677e-002 | - |
1/64 | 1.1104e-003 | 4.00 | 5.8545e-003 | 1.99 |
1/128 | 1.0692e-004 | 4.00 | 2.9303e-003 | 2.00 |
1/256 | 6.6840e-006 | 4.00 | 1.4660e-003 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 2.2208e-003 | - | 3.8223e-004 | - |
1/64 | 1/642 | 5.6284e-004 | 3.95 | 1.0418e-004 | 3.67 |
1/128 | 1/1282 | 1.4169e-004 | 3.97 | 2.7176e-005 | 3.83 |
1/256 | 1/2562 | 3.5544e-005 | 3.99 | 6.9401e-006 | 3.92 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 5.1907e-002 | - | 1.0436e-002 | - |
1/64 | 1/64 | 2.7983e-002 | 1.85 | 5.7010e-003 | 1.83 |
1/128 | 1/128 | 1.4518e-002 | 1.93 | 2.9780e-003 | 1.91 |
1/256 | 1/256 | 7.3924e-003 | 1.96 | 1.5219e-003 | 1.96 |
This work focuses on a heat conduction problem with nonlocal boundary conditions. We develop an implicit Euler scheme and demonstrate that it achieves asymptotic optimal order with the DFT. Furthermore, we introduce two approximation formulas that exhibit superapproximation for first-order partial derivatives along the x and y directions of the exact solution, respectively. In the future, we plan to extend this work to other difference schemes for parabolic problems with nonlocal boundary conditions, such as the explicit Euler scheme, the Crank-Nicolson scheme, and other schemes. Additionally, we aim to consider heat conduction problems with different nonlocal boundary conditions.
Liping Zhou: Conceptualization, methodology, formal analysis, writing-original draft, validation; Yumei Yan: Editing, software; Ying Liu: Writing-review and editing. All authors contributed equally to the manuscript. All authors have read and approved the final version of the manuscript for publication.
This work is partially supported by the National Natural Science Foundation of China (No.12101224), the Hunan Provincial Natural Science Foundation of China (No. 2022JJ30271, 2024JJ7203) and the Key Project of Hunan Provincial Education Department of China (No. 23A0577).
The authors declare no conflicts of interest.
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h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.4377e-003 | - | 1.2941e-001 | - |
1/64 | 1.1104e-003 | 4.00 | 6.5173e-002 | 1.99 |
1/128 | 2.7768e-004 | 4.00 | 3.2701e-002 | 1.99 |
1/256 | 6.9427e-005 | 4.00 | 1.6379e-002 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 1.7481e-002 | - | 7.0681e-003 | - |
1/64 | 1/642 | 4.4901e-003 | 3.89 | 1.8960e-003 | 3.73 |
1/128 | 1/1282 | 1.1379e-003 | 3.95 | 5.0110e-004 | 3.78 |
1/256 | 1/2562 | 2.8640e-004 | 3.97 | 1.3025e-004 | 3.85 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 6.1798e-001 | - | 1.7722e-001 | - |
1/64 | 1/64 | 3.3247e-001 | 1.86 | 1.0037e-001 | 1.77 |
1/128 | 1/128 | 1.7242e-001 | 1.93 | 5.3329e-002 | 1.88 |
1/256 | 1/256 | 8.7795e-002 | 1.96 | 2.7478e-002 | 1.94 |
h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.2726e-004 | - | 1.1677e-002 | - |
1/64 | 1.1104e-003 | 4.00 | 5.8545e-003 | 1.99 |
1/128 | 1.0692e-004 | 4.00 | 2.9303e-003 | 2.00 |
1/256 | 6.6840e-006 | 4.00 | 1.4660e-003 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 2.2208e-003 | - | 3.8223e-004 | - |
1/64 | 1/642 | 5.6284e-004 | 3.95 | 1.0418e-004 | 3.67 |
1/128 | 1/1282 | 1.4169e-004 | 3.97 | 2.7176e-005 | 3.83 |
1/256 | 1/2562 | 3.5544e-005 | 3.99 | 6.9401e-006 | 3.92 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 5.1907e-002 | - | 1.0436e-002 | - |
1/64 | 1/64 | 2.7983e-002 | 1.85 | 5.7010e-003 | 1.83 |
1/128 | 1/128 | 1.4518e-002 | 1.93 | 2.9780e-003 | 1.91 |
1/256 | 1/256 | 7.3924e-003 | 1.96 | 1.5219e-003 | 1.96 |
h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.4377e-003 | - | 1.2941e-001 | - |
1/64 | 1.1104e-003 | 4.00 | 6.5173e-002 | 1.99 |
1/128 | 2.7768e-004 | 4.00 | 3.2701e-002 | 1.99 |
1/256 | 6.9427e-005 | 4.00 | 1.6379e-002 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 1.7481e-002 | - | 7.0681e-003 | - |
1/64 | 1/642 | 4.4901e-003 | 3.89 | 1.8960e-003 | 3.73 |
1/128 | 1/1282 | 1.1379e-003 | 3.95 | 5.0110e-004 | 3.78 |
1/256 | 1/2562 | 2.8640e-004 | 3.97 | 1.3025e-004 | 3.85 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 6.1798e-001 | - | 1.7722e-001 | - |
1/64 | 1/64 | 3.3247e-001 | 1.86 | 1.0037e-001 | 1.77 |
1/128 | 1/128 | 1.7242e-001 | 1.93 | 5.3329e-002 | 1.88 |
1/256 | 1/256 | 8.7795e-002 | 1.96 | 2.7478e-002 | 1.94 |
h | τ=h2 | τ=h | ||
‖U−u‖∞ | ratio | ‖U−u‖∞ | ratio | |
1/32 | 4.2726e-004 | - | 1.1677e-002 | - |
1/64 | 1.1104e-003 | 4.00 | 5.8545e-003 | 1.99 |
1/128 | 1.0692e-004 | 4.00 | 2.9303e-003 | 2.00 |
1/256 | 6.6840e-006 | 4.00 | 1.4660e-003 | 2.00 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/322 | 2.2208e-003 | - | 3.8223e-004 | - |
1/64 | 1/642 | 5.6284e-004 | 3.95 | 1.0418e-004 | 3.67 |
1/128 | 1/1282 | 1.4169e-004 | 3.97 | 2.7176e-005 | 3.83 |
1/256 | 1/2562 | 3.5544e-005 | 3.99 | 6.9401e-006 | 3.92 |
h | τ | ‖Ux−ux‖∞ | ratio | ‖Uy−uy‖∞ | ratio |
1/32 | 1/32 | 5.1907e-002 | - | 1.0436e-002 | - |
1/64 | 1/64 | 2.7983e-002 | 1.85 | 5.7010e-003 | 1.83 |
1/128 | 1/128 | 1.4518e-002 | 1.93 | 2.9780e-003 | 1.91 |
1/256 | 1/256 | 7.3924e-003 | 1.96 | 1.5219e-003 | 1.96 |