# Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Authors: Darae Jeong; Yibao Li; Chaeyoung Lee; Junxiang Yang; Yongho Choi; Junseok Kim
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.


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## Body

## 1. Introduction

Many numerical schemes for differential equations have been developed and tested in terms of convergence rates. If an analytic solution is available, then we can use it as a reference solution for the convergence test. Otherwise, we can use a reference solution obtained from a numerical solution with very fine space and time step sizes. Some scientific journal considers manuscripts only if accuracy and convergence of numerical solutions are established by discussion of results on multiple grids. Recently, there have been many research studies on the second-order convergence schemes for parabolic-type partial differential equations [1–12]. To demonstrate the second-order convergence, some authors [1–3] showed the convergence by refining the spatial and temporal steps at the same time; some authors in [4–10] showed the convergence by refining the spatial and temporal steps separately.In this work, we will show that we may have the second-order convergence even though the numerical scheme is the first-order accurate in time and the second-order accurate in space if we refine the spatial and temporal steps at the same time. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately. We validate these claims from the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the convergence rates of the numerical schemes for the heat equation, the Allen–Cahn (AC) equation, and the Cahn–Hilliard (CH) equation, which have been studied theoretically by many researchers in recent years [14–20].The first equation is the heat equation onΩ=(0,2π):(1)∂ϕx,t∂t=∂2ϕx,t∂x2fort>0with the zero Neumann boundary condition ϕx(0,t)=ϕx(2π,t)=0.The second equation is the AC equation [22, 23] on Ω=(0,1):(2)∂ϕx,t∂t=F′ϕx,tϵ2+Δϕx,tfort>0with the periodic boundary condition ϕ(0,t)=ϕ(1,t). The phase-field ϕ(x,t) is the difference between the concentrations of the two mixtures’ components, F(ϕ)=0.25(ϕ2-1)2, and ϵ is a positive constant. The AC equation has a wide range of applications such as mean curvature flows [24–26], two-phase incompressible fluids [27], complex dynamics of dendritic growth [28, 29], image inpainting [30], and image segmentation [31, 32].The third equation is the CH equation [33] on Ω=(0,1): (3)∂ϕ∂tx,t=Δμx,tfort>0,(4)μx,t=F′ϕx,t-ϵ2Δϕx,t, with the periodic boundary condition (5)ϕ0,t=ϕ1,t,μ0,t=μ1,t. This CH equation is widely used in applications such as phase separation [34], topology optimization [35], multiphase incompressible fluid flows [36–39], image inpainting [40], surface reconstruction [41], diblock copolymer [42], tumor growth simulation [43], and microstructures with elastic inhomogeneity [44].Some studies suggest a mathematical model as a way to reproduce the experiment and present a numerical solution of it, as shown in Figures1 and 2. Figure 1 shows the experimental and numerical pattern formations of binary mixture [13].Figure 1
Phase separation in a binary mixture by (a) experiment and (b) numerical tests. Adapted from Voit et al. [13] with the permission of American Physical Society.
(a) (b)Figure 2
Formation of block copolymer by numerical (first row) and numerical (second row) tests. Adapted from Horvat et al. [21] with the permission of American Chemical Society.
(a) (b) (c) (d) (e) (f) (g) (h)Other example [21] is the formation of block copolymer; the numerical and experimental results are shown in first and second rows in Figure 2, respectively.Along with the study of mathematical model, various numerical schemes have been developed as a way to solve this accurately. At this time, a convergence test can be performed as a method for verifying the accuracy of such a numerical solution, which must be performed correctly.Therefore, we present the right method validating the accuracy order of the numerical solution for parabolic-type equations with various benchmark tests.This paper is structured in the following manner. In Section2, we describe the numerical solution algorithms for the three equations. In Section 3, we present several numerical results. Then, in Section 4, we conclude.


## 2. Numerical Solution

In this section, we present the numerical solutions for the three equations by a finite difference method [45].


### 2.1. Heat Equation

First, we consider the heat equation. LetΩ=(0,2π) be discretized by using a uniform grid with h=2π/Nx, where Nx is the number of subintervals (see Figure 3).Figure 3
Uniform grid with a spatial step sizeh.Letϕin be approximation of ϕ(xi,tn), where xi=(i-0.5)h, tn=nΔt, and Δt is the temporal step size. Now, we apply a θ-method to the heat (1) as follows:(6)ϕin+1-ϕinΔt=1-θΔhϕin+θΔhϕin+1,for1≤i≤Nx,where Δhϕin=(ϕi-1n-2ϕin+ϕi+1n)/h2 and 0≤θ≤1. For the homogeneous Neumann boundary condition, we set ϕ0n=ϕ1n and ϕNx+1n=ϕNxn for all n=0,1,…. If θ=0.5, then (6) becomes the Crank–Nicolson (CN) scheme [46] and the convergence rate is O(Δt2)+O(h2). If θ=1, then (6) becomes a fully implicit scheme with the convergence rate, O(Δt)+O(h2). Here, (6) is solved by using the Thomas algorithm [45]. Let the error be defined as eNxNt=(e1Nt,e2Nt,…,eNxNt), where eiNt=ϕiNt-ϕ(xi,tNt) for i=1,…,Nx and let ‖eNxNt‖2=1/Nx∑i=1Nx‍eiNt2.


### 2.2. AC Equation

We consider the numerical solution for the AC equation. Letϕin be approximation of ϕ(xi,tn), where xi=ih and h=1/Nx. We discretize (2) by applying the θ-method as follows:(7)ϕin+1-ϕinΔt=1-θ-F′ϕinϵ2+Δhϕin+θ-F′ϕin+1ϵ2+Δhϕin+1,where 0≤θ≤1 and 0≤i≤Nx. For the periodic boundary condition, we set ϕ0n=ϕNxn for all n=0,1,…. Here, we use a multigrid algorithm [47–49] to solve the discrete (7).


### 2.3. CH Equation

In this problem, we apply the unconditionally gradient stable method [50] to (3) and (4):(8)ϕin+1-ϕinΔt=Δhμin+1,for0≤i≤Nx,(9)μin+1=ϕin+13-ϕin-ϵ2Δϕin+1with the periodic boundary condition. Note that this scheme is first-order in time and second-order in space. To solve the discrete equation (8) and (9), we also use a multigrid algorithm [47–49].


## 2.1. Heat Equation

First, we consider the heat equation. LetΩ=(0,2π) be discretized by using a uniform grid with h=2π/Nx, where Nx is the number of subintervals (see Figure 3).Figure 3
Uniform grid with a spatial step sizeh.Letϕin be approximation of ϕ(xi,tn), where xi=(i-0.5)h, tn=nΔt, and Δt is the temporal step size. Now, we apply a θ-method to the heat (1) as follows:(6)ϕin+1-ϕinΔt=1-θΔhϕin+θΔhϕin+1,for1≤i≤Nx,where Δhϕin=(ϕi-1n-2ϕin+ϕi+1n)/h2 and 0≤θ≤1. For the homogeneous Neumann boundary condition, we set ϕ0n=ϕ1n and ϕNx+1n=ϕNxn for all n=0,1,…. If θ=0.5, then (6) becomes the Crank–Nicolson (CN) scheme [46] and the convergence rate is O(Δt2)+O(h2). If θ=1, then (6) becomes a fully implicit scheme with the convergence rate, O(Δt)+O(h2). Here, (6) is solved by using the Thomas algorithm [45]. Let the error be defined as eNxNt=(e1Nt,e2Nt,…,eNxNt), where eiNt=ϕiNt-ϕ(xi,tNt) for i=1,…,Nx and let ‖eNxNt‖2=1/Nx∑i=1Nx‍eiNt2.


## 2.2. AC Equation

We consider the numerical solution for the AC equation. Letϕin be approximation of ϕ(xi,tn), where xi=ih and h=1/Nx. We discretize (2) by applying the θ-method as follows:(7)ϕin+1-ϕinΔt=1-θ-F′ϕinϵ2+Δhϕin+θ-F′ϕin+1ϵ2+Δhϕin+1,where 0≤θ≤1 and 0≤i≤Nx. For the periodic boundary condition, we set ϕ0n=ϕNxn for all n=0,1,…. Here, we use a multigrid algorithm [47–49] to solve the discrete (7).


## 2.3. CH Equation

In this problem, we apply the unconditionally gradient stable method [50] to (3) and (4):(8)ϕin+1-ϕinΔt=Δhμin+1,for0≤i≤Nx,(9)μin+1=ϕin+13-ϕin-ϵ2Δϕin+1with the periodic boundary condition. Note that this scheme is first-order in time and second-order in space. To solve the discrete equation (8) and (9), we also use a multigrid algorithm [47–49].


## 3. Numerical Experiments

### 3.1. Heat Equation

For numercial test, the initial state is given byϕ(x,0)=cos⁡(x) on Ω=(0,2π). Therefore, the closed-form solution of (1) is ϕ(x,t)=e-tcos⁡(x). Now, we compute the discrete l2-norm error at T=0.1.


#### 3.1.1. Convergence Test for the CN Scheme

Table1 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. Here, the temporal convergence rate is defined as log2‖eNxNt‖2/‖eNx2Nt‖2. The boxed rates are the numbers greater than or equal to 1.9. As we refine the space grid size, we have more second-order temporal convergence results.Table 1
l 2-norm error and temporal convergence rates for the CN scheme with various Δt at T=0.1.
N t ∖ N x 100 200 400 800 1600 1 3.23e-5 4.81e-5 5.21e-5 5.31e-5 5.33e-5 rate 2.06 2.58 2.12 2.03 2.01 2 7.73e-6 8.07e-6 1.20e-5 1.30e-5 1.33e-5 rate -1.20 2.06 2.57 2.11 2.03 4 1.77e-5 1.93e-6 2.02e-6 3.00e-6 3.25e-6 rate -0.19 -1.20 2.06 2.57 2.11 8 2.02e-5 4.43e-6 4.82e-7 5.04e-7 7.51e-7 rate -0.04 -0.19 -1.20 2.06 2.57 16 2.08e-5 5.05e-6 1.11e-6 1.21e-7 1.26e-7 rate -0.01 -0.04 -0.19 -1.20 2.06 32 2.10e-5 5.21e-6 1.26e-6 2.77e-7 3.02e-8 rate -0.00 -0.01 -0.04 -0.19 -1.20 64 2.10e-5 5.25e-6 1.30e-6 3.16e-7 6.92e-8 rate -0.00 -0.00 -0.01 -0.04 -0.19 128 2.10e-5 5.26e-6 1.31e-6 3.26e-7 7.90e-8 rate -0.00 -0.00 -0.00 -0.01 -0.04 256 2.10e-5 5.26e-6 1.31e-6 3.28e-7 8.14e-8 rate -0.00 -0.00 -0.00 -0.00 -0.01 512 2.10e-5 5.26e-6 1.32e-6 3.29e-7 8.20e-8Table2 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. The spatial convergence rate is defined as log2‖eNxNt‖2/‖e2NxNt‖2. As we refine the time grid size, we have more second-order spatial convergence results.Table 2
l 2-norm error and spatial convergence rates for the CN scheme with various h at T=0.1.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5 -0.58 4.81e-5 -0.11 5.21e-5 -0.03 5.31e-5 -0.01 5.33e-5 2 7.73e-6 -0.06 8.07e-6 -0.57 1.20e-5 -0.11 1.30e-5 -0.03 1.33e-5 4 1.77e-5 3.20 1.93e-6 -0.06 2.02e-6 -0.57 3.00e-6 -0.11 3.25e-6 8 2.02e-5 2.19 4.43e-6 3.20 4.82e-7 -0.06 5.04e-7 -0.57 7.51e-7 16 2.08e-5 2.04 5.05e-6 2.19 1.11e-6 3.20 1.21e-7 -0.06 1.26e-7 32 2.10e-5 2.01 5.21e-6 2.04 1.26e-6 2.19 2.77e-7 3.20 3.02e-8 64 2.10e-5 2.00 5.25e-6 2.01 1.30e-6 2.04 3.16e-7 2.19 6.92e-8 128 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.01 3.26e-7 2.04 7.90e-8 256 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.00 3.28e-7 2.01 8.14e-8 512 2.10e-5 2.00 5.26e-6 2.00 1.32e-6 2.00 3.29e-7 2.00 8.20e-8Table3 lists the l2-norm error and convergence rates for the CN scheme with various h=2π/Nx and Δt=T/Nt. The convergence rate is defined as log2‖eNxNt‖2/‖e2Nx2Nt‖2. We have second-order convergence results for all the cases. We have a sequence of the second-order convergence (diagonal sequence in Table 3) regardless of starting pair, (Nx,Nt). We can find some examples of this approach. In [1], the refinement path was taken to be Δt=0.2h/2 to show the second-order convergence of the proposed numerical scheme for the Cahn–Hilliard–Navier–Stokes equation. In [2], Δt=h was taken to show the second-order convergence for solving space fractional diffusion equations. In [3], Δt=h was used to show the second-order convergence rate for a two-sided space-fractional diffusion equation with variable coefficients.Table 3
l 2-norm error and convergence rates for the CN scheme with various Δt and h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5  4.81e-5  5.21e-5  5.31e-5  5.33e-5 rate  2.00  2.00  2.00  2.00 2 7.73e-6  8.07e-6  1.20e-5  1.30e-5  1.33e-5 rate  2.00  2.00  2.00  2.00 4 1.77e-5  1.93e-6  2.02e-6  3.00e-6  3.25e-6 rate  2.00  2.00  2.00  2.00 8 2.02e-5  4.43e-6  4.82e-7  5.04e-7  7.51e-7 rate  2.00  2.00  2.00  2.00 16 2.08e-5  5.05e-6  1.11e-6  1.21e-7  1.26e-7 rate  2.00  2.00  2.00  2.00 32 2.10e-5  5.21e-6  1.26e-6  2.77e-7  3.02e-8 rate  2.00  2.00  2.00  2.00 64 2.10e-5  5.25e-6  1.30e-6  3.16e-7  6.92e-8 rate  2.00  2.00  2.00  2.00 128 2.10e-5  5.26e-6  1.31e-6  3.26e-7  7.90e-8 rate  2.00  2.00  2.00  2.00 256 2.10e-5  5.26e-6  1.31e-6  3.28e-7  8.14e-8 rate  2.00  2.00  2.00  2.00 512 2.10e-5  5.26e-6  1.32e-6  3.29e-7  8.20e-8We have the following relation between errors:(10)log2eNxNt2e2Nx2Nt2=log2eNxNt2eNx2Nt2+log2eNx2Nt2e2Nx2Nt2=log2eNxNt2e2NxNt2+log2e2NxNt2e2Nx2Nt2.From the convergence results in Tables1, 2, and 3, we can confirm this relation. This convergence relation implies that if we refine both the time and space steps, then we may have the second-order convergence even though one of two convergence rates is not second-order accurate. Interestingly, the convergence relation implies that we do not have the second-order convergence results for both the spatial and temporal convergence rates at the same starting pair, (Nx,Nt).


#### 3.1.2. Convergence Test for the Fully Implicit Scheme

Table4 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. As we refine the space grid size, we have more first-order temporal convergence results.Table 4
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt.
N t ∖ N x 100 200 400 800 1600 512 2.73e-5 1.15e-5 7.56e-6 6.58e-6 6.33e-6 rate 0.18 0.46 0.77 0.93 0.98 1024 2.42e-5 8.39e-6 4.44e-6 3.45e-6 3.21e-6 rate 0.10 0.30 0.63 0.87 0.96 2048 2.26e-5 6.82e-6 2.88e-6 1.89e-6 1.64e-6 rate 0.05 0.18 0.46 0.77 0.93 4096 2.18e-5 6.04e-6 2.10e-6 1.11e-6 8.63e-7 rate 0.03 0.10 0.30 0.63 0.87 8192 2.14e-5 5.65e-6 1.71e-6 7.19e-7 4.73e-7 rate 0.01 0.05 0.18 0.46 0.77 16384 2.12e-5 5.46e-6 1.51e-6 5.24e-7 2.77e-7 rate 0.01 0.03 0.10 0.30 0.63 32768 2.11e-5 5.36e-6 1.41e-6 4.27e-7 1.80e-7 rate 0.00 0.01 0.05 0.18 0.46 65536 2.11e-5 5.31e-6 1.36e-6 3.78e-7 1.31e-7 rate 0.00 0.01 0.03 0.10 0.30 131072 2.11e-5 5.29e-6 1.34e-6 3.53e-7 1.07e-7 rate 0.00 0.00 0.01 0.05 0.18 262144 2.11e-5 5.27e-6 1.33e-6 3.41e-7 9.44e-8Table5 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. As we refine the time step size, we have more second-order spatial convergence results.Table 5
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5 1.25 1.15e-5 0.61 7.56e-6 0.20 6.58e-6 0.06 6.33e-6 1024 2.42e-5 1.53 8.39e-6 0.92 4.44e-6 0.36 3.45e-6 0.11 3.21e-6 2048 2.26e-5 1.73 6.82e-6 1.25 2.88e-6 0.61 1.89e-6 0.20 1.64e-6 4096 2.18e-5 1.85 6.04e-6 1.53 2.10e-6 0.92 1.11e-6 0.36 8.63e-7 8192 2.14e-5 1.92 5.65e-6 1.73 1.71e-6 1.25 7.19e-7 0.61 4.73e-7 16384 2.12e-5 1.96 5.46e-6 1.85 1.51e-6 1.53 5.24e-7 0.92 2.77e-7 32768 2.11e-5 1.98 5.36e-6 1.92 1.41e-6 1.73 4.27e-7 1.25 1.80e-7 65536 2.11e-5 1.99 5.31e-6 1.96 1.36e-6 1.85 3.78e-7 1.53 1.31e-7 131072 2.11e-5 1.99 5.29e-6 1.98 1.34e-6 1.92 3.53e-7 1.73 1.07e-7 262144 2.11e-5 2.00 5.27e-6 1.99 1.33e-6 1.96 3.41e-7 1.85 9.44e-8Table6 lists the l2-norm error and convergence rates for the fully implicit scheme with various h=2π/Nx and Δt=T/Nt. We have a range of convergence results from the first-order to the second-order accuracy. In the lower left triangular region in Table 6, the magnitude of the spatial discretization error dominates that of the temporal discretization error. Therefore, we have the second-order convergence. In the upper right triangular region in Table 6, the magnitude of the temporal discretization error dominates that of the spatial discretization error. Therefore, we have the first-order convergence. This result implies that if we refine both the time and space steps, then we may have the second-order convergence even though the fully implicit scheme is first-order accurate in temporal discretization.Table 6
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h.
N t \ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5  1.15e-5  7.56e-6  6.58e-6  6.33e-6 rate  1.70  1.37  1.13  1.04 1024 2.42e-5  8.39e-6  4.44e-6  3.45e-6  3.21e-6 rate  1.82  1.54  1.23  1.07 2048 2.26e-5  6.82e-6  2.88e-6  1.89e-6  1.64e-6 rate  1.90  1.70  1.37  1.13 4096 2.18e-5  6.04e-6  2.10e-6  1.11e-6  8.63e-7 rate  1.95  1.82  1.54  1.23 8192 2.14e-5  5.65e-6  1.71e-6  7.19e-7  4.73e-7 rate  1.97  1.90  1.70  1.37 16384 2.12e-5  5.46e-6  1.51e-6  5.24e-7  2.77e-7 rate  1.99  1.95  1.82  1.54 32768 2.11e-5  5.36e-6  1.41e-6  4.27e-7  1.80e-7 rate  1.99  1.97  1.90  1.70 65536 2.11e-5  5.31e-6  1.36e-6  3.78e-7  1.31e-7 rate  2.00  1.99  1.95  1.82 131072 2.11e-5  5.29e-6  1.34e-6  3.53e-7  1.07e-7 rate  2.00  1.99  1.97  1.90 262144 2.11e-5  5.27e-6  1.33e-6  3.41e-7  9.44e-8


### 3.2. The Allen–Cahn Equation

The initial condition isϕ(x)=0.2cos⁡(2πx) on Ω=(0,1). We use T = 1.0e-5, and ϵ=0.0075. Because there is no analytic solution for (7), we consider a reference solution. We define the reference solution ϕref as the numerical solution with very fine space and time steps, (Δtref, href) = (3.81e-12, 1/2048).


#### 3.2.1. Convergence Test for the CN Scheme

Table7 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt with a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the second-order temporal convergence result.Table 7
l 2-norm error and temporal convergence rates for the CN scheme with various Δt. (Δt, h)=(1.25e-7, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 7.13e-9 2.00 1.78e-9 2.00 4.46e-10 2.00 1.11e-10Table8 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=1/Nx with a fixed time step size Δt=T/2621440. As we refine the space grid size, i.e., Nx=8,16,32 and 64, we have the second-order spatial convergence result.Table 8
l 2-norm error and spatial convergence rates for the CN scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8Tables9 and 10 list the l2-norm error and spatial and temporal convergence rates for the CN scheme with various h=1/Nx and Δt=T/Nt. The results generate the second-order convergence for both the cases, i.e., (Δt, h)=(1.25e-7, 1/64) and (Δt, h)=(6.10e-11, 1/8).Table 9
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 6.03e-8 2.01 1.50e-8 2.02 3.69e-9 2.09 8.68e-10Table 10
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8


#### 3.2.2. Convergence Test for the Fully Implicit Scheme

Table11 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the first-order temporal convergence result.Table 11
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt. (Δt, h)=(1.25e-07, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 2.82e-5 1.01 1.40e-5 1.01 6.96e-6 1.02 3.42e-6Table12 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=1/2048 with a fixed time step size Δt=T/Nt. As we refine the space grid size, i.e., Nx=8,16,32, and 64, we have the second-order spatial convergence result.Table 12
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.46e-6 1.99 8.69e-7 1.97 2.22e-7 1.86 6.10e-8Tables13 and 14 list the l2-norm error and spatial and temporal convergence rates for the fully implicit scheme with various h=1/Nx and Δt=T/Nt, respectively.Table 13
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 2.85e-5 1.01 1.42e-5 1.00 7.08e-6 1.00 3.54e-6Table 14
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.47e-6 2.00 8.67e-7 2.00 2.17e-7 2.00 5.41e-8Table13 shows that the convergence rate appears to be only first-order with Nt=80 and Nx=64. On the other hand, Table 14 shows that the convergence rate appears to be second-order with Nt=163840 and Nx=8. These results demonstrate that even if the numerical scheme is only first-order in time and second-order in space, we may have the second-order convergence with some refinement combination.


### 3.3. The Cahn–Hilliard Equation

The initial condition isϕ(x)=0.1cos⁡(2πx) on Ω=(0,1). Table 15 lists the l2-norm error and temporal convergence rates for the unconditionally stable scheme with various Δt=T/Nt with a fixed space step size h=1/Nx. We use Nx=1024, T = 2.40e-3, and ϵ=0.02. We define the reference solution with fine time step Δt = 9.54e-6. We find that the convergence rates are all first-order.Table 15
l 2-norm error and temporal convergence rates for the unconditionally stable numerical scheme with various Δt. (Δt, h)=(3.05e-4, 1/1024) and (Δtref, href</caption>) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) l 2-error 4.71e-5 1.05 2.27e-5 1.11 1.05e-5Table16 lists the l2-norm error and spatial convergence rates with various h=1/Nx with a fixed time step size Δt=T/Nt. We use Δt = 9.54e-6, T = 2.40e-3, and ϵ=0.02. We define the reference solution with fine space step h=1/1024. We can find that the convergence rates are second-order.Table 16
l 2-norm error and spatial convergence rates for the unconditionally stable numerical scheme with various h. (Δt, h)=(9.54e-6, 1/16) and (Δtref, href</caption>) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) l 2-error 9.90e-5 1.98 2.52e-5 1.96 6.46e-6Table17 lists the l2-norm error and spatial and temporal convergence rates with various h=1/Nx and Δt. We use T = 2.40e-3 and ϵ=0.02. We can find that the convergence rates are first-order.Table 17
l 2-norm error and convergence rates for the unconditionally stable numerical scheme with various Δt and h. (Δt, h)=(1.20e-3, 1/32) and (Δtref, href) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) l 2-error 2.06e-4 1.06 9.87e-5 1.00 4.93e-5Table18 lists the l2-norm error and spatial and temporal convergence rates with various h=1/Nx and Δt. We use T = 2.40e-3 and ϵ=0.02. We can find that the convergence rates are second-order.Table 18
l 2-norm error and convergence rates for the unconditionally stable numerical scheme with various Δt and h. (Δt, h)=(7.63e-5, 1/8) and (Δtref, href) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) l 2-error 3.88e-4 1.91 1.03e-4 1.95 2.67e-5The results in Tables17 and 18 show that we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes in the case of the first-order in time and the second-order in space scheme.


## 3.1. Heat Equation

For numercial test, the initial state is given byϕ(x,0)=cos⁡(x) on Ω=(0,2π). Therefore, the closed-form solution of (1) is ϕ(x,t)=e-tcos⁡(x). Now, we compute the discrete l2-norm error at T=0.1.


### 3.1.1. Convergence Test for the CN Scheme

Table1 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. Here, the temporal convergence rate is defined as log2‖eNxNt‖2/‖eNx2Nt‖2. The boxed rates are the numbers greater than or equal to 1.9. As we refine the space grid size, we have more second-order temporal convergence results.Table 1
l 2-norm error and temporal convergence rates for the CN scheme with various Δt at T=0.1.
N t ∖ N x 100 200 400 800 1600 1 3.23e-5 4.81e-5 5.21e-5 5.31e-5 5.33e-5 rate 2.06 2.58 2.12 2.03 2.01 2 7.73e-6 8.07e-6 1.20e-5 1.30e-5 1.33e-5 rate -1.20 2.06 2.57 2.11 2.03 4 1.77e-5 1.93e-6 2.02e-6 3.00e-6 3.25e-6 rate -0.19 -1.20 2.06 2.57 2.11 8 2.02e-5 4.43e-6 4.82e-7 5.04e-7 7.51e-7 rate -0.04 -0.19 -1.20 2.06 2.57 16 2.08e-5 5.05e-6 1.11e-6 1.21e-7 1.26e-7 rate -0.01 -0.04 -0.19 -1.20 2.06 32 2.10e-5 5.21e-6 1.26e-6 2.77e-7 3.02e-8 rate -0.00 -0.01 -0.04 -0.19 -1.20 64 2.10e-5 5.25e-6 1.30e-6 3.16e-7 6.92e-8 rate -0.00 -0.00 -0.01 -0.04 -0.19 128 2.10e-5 5.26e-6 1.31e-6 3.26e-7 7.90e-8 rate -0.00 -0.00 -0.00 -0.01 -0.04 256 2.10e-5 5.26e-6 1.31e-6 3.28e-7 8.14e-8 rate -0.00 -0.00 -0.00 -0.00 -0.01 512 2.10e-5 5.26e-6 1.32e-6 3.29e-7 8.20e-8Table2 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. The spatial convergence rate is defined as log2‖eNxNt‖2/‖e2NxNt‖2. As we refine the time grid size, we have more second-order spatial convergence results.Table 2
l 2-norm error and spatial convergence rates for the CN scheme with various h at T=0.1.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5 -0.58 4.81e-5 -0.11 5.21e-5 -0.03 5.31e-5 -0.01 5.33e-5 2 7.73e-6 -0.06 8.07e-6 -0.57 1.20e-5 -0.11 1.30e-5 -0.03 1.33e-5 4 1.77e-5 3.20 1.93e-6 -0.06 2.02e-6 -0.57 3.00e-6 -0.11 3.25e-6 8 2.02e-5 2.19 4.43e-6 3.20 4.82e-7 -0.06 5.04e-7 -0.57 7.51e-7 16 2.08e-5 2.04 5.05e-6 2.19 1.11e-6 3.20 1.21e-7 -0.06 1.26e-7 32 2.10e-5 2.01 5.21e-6 2.04 1.26e-6 2.19 2.77e-7 3.20 3.02e-8 64 2.10e-5 2.00 5.25e-6 2.01 1.30e-6 2.04 3.16e-7 2.19 6.92e-8 128 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.01 3.26e-7 2.04 7.90e-8 256 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.00 3.28e-7 2.01 8.14e-8 512 2.10e-5 2.00 5.26e-6 2.00 1.32e-6 2.00 3.29e-7 2.00 8.20e-8Table3 lists the l2-norm error and convergence rates for the CN scheme with various h=2π/Nx and Δt=T/Nt. The convergence rate is defined as log2‖eNxNt‖2/‖e2Nx2Nt‖2. We have second-order convergence results for all the cases. We have a sequence of the second-order convergence (diagonal sequence in Table 3) regardless of starting pair, (Nx,Nt). We can find some examples of this approach. In [1], the refinement path was taken to be Δt=0.2h/2 to show the second-order convergence of the proposed numerical scheme for the Cahn–Hilliard–Navier–Stokes equation. In [2], Δt=h was taken to show the second-order convergence for solving space fractional diffusion equations. In [3], Δt=h was used to show the second-order convergence rate for a two-sided space-fractional diffusion equation with variable coefficients.Table 3
l 2-norm error and convergence rates for the CN scheme with various Δt and h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5  4.81e-5  5.21e-5  5.31e-5  5.33e-5 rate  2.00  2.00  2.00  2.00 2 7.73e-6  8.07e-6  1.20e-5  1.30e-5  1.33e-5 rate  2.00  2.00  2.00  2.00 4 1.77e-5  1.93e-6  2.02e-6  3.00e-6  3.25e-6 rate  2.00  2.00  2.00  2.00 8 2.02e-5  4.43e-6  4.82e-7  5.04e-7  7.51e-7 rate  2.00  2.00  2.00  2.00 16 2.08e-5  5.05e-6  1.11e-6  1.21e-7  1.26e-7 rate  2.00  2.00  2.00  2.00 32 2.10e-5  5.21e-6  1.26e-6  2.77e-7  3.02e-8 rate  2.00  2.00  2.00  2.00 64 2.10e-5  5.25e-6  1.30e-6  3.16e-7  6.92e-8 rate  2.00  2.00  2.00  2.00 128 2.10e-5  5.26e-6  1.31e-6  3.26e-7  7.90e-8 rate  2.00  2.00  2.00  2.00 256 2.10e-5  5.26e-6  1.31e-6  3.28e-7  8.14e-8 rate  2.00  2.00  2.00  2.00 512 2.10e-5  5.26e-6  1.32e-6  3.29e-7  8.20e-8We have the following relation between errors:(10)log2eNxNt2e2Nx2Nt2=log2eNxNt2eNx2Nt2+log2eNx2Nt2e2Nx2Nt2=log2eNxNt2e2NxNt2+log2e2NxNt2e2Nx2Nt2.From the convergence results in Tables1, 2, and 3, we can confirm this relation. This convergence relation implies that if we refine both the time and space steps, then we may have the second-order convergence even though one of two convergence rates is not second-order accurate. Interestingly, the convergence relation implies that we do not have the second-order convergence results for both the spatial and temporal convergence rates at the same starting pair, (Nx,Nt).


### 3.1.2. Convergence Test for the Fully Implicit Scheme

Table4 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. As we refine the space grid size, we have more first-order temporal convergence results.Table 4
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt.
N t ∖ N x 100 200 400 800 1600 512 2.73e-5 1.15e-5 7.56e-6 6.58e-6 6.33e-6 rate 0.18 0.46 0.77 0.93 0.98 1024 2.42e-5 8.39e-6 4.44e-6 3.45e-6 3.21e-6 rate 0.10 0.30 0.63 0.87 0.96 2048 2.26e-5 6.82e-6 2.88e-6 1.89e-6 1.64e-6 rate 0.05 0.18 0.46 0.77 0.93 4096 2.18e-5 6.04e-6 2.10e-6 1.11e-6 8.63e-7 rate 0.03 0.10 0.30 0.63 0.87 8192 2.14e-5 5.65e-6 1.71e-6 7.19e-7 4.73e-7 rate 0.01 0.05 0.18 0.46 0.77 16384 2.12e-5 5.46e-6 1.51e-6 5.24e-7 2.77e-7 rate 0.01 0.03 0.10 0.30 0.63 32768 2.11e-5 5.36e-6 1.41e-6 4.27e-7 1.80e-7 rate 0.00 0.01 0.05 0.18 0.46 65536 2.11e-5 5.31e-6 1.36e-6 3.78e-7 1.31e-7 rate 0.00 0.01 0.03 0.10 0.30 131072 2.11e-5 5.29e-6 1.34e-6 3.53e-7 1.07e-7 rate 0.00 0.00 0.01 0.05 0.18 262144 2.11e-5 5.27e-6 1.33e-6 3.41e-7 9.44e-8Table5 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. As we refine the time step size, we have more second-order spatial convergence results.Table 5
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5 1.25 1.15e-5 0.61 7.56e-6 0.20 6.58e-6 0.06 6.33e-6 1024 2.42e-5 1.53 8.39e-6 0.92 4.44e-6 0.36 3.45e-6 0.11 3.21e-6 2048 2.26e-5 1.73 6.82e-6 1.25 2.88e-6 0.61 1.89e-6 0.20 1.64e-6 4096 2.18e-5 1.85 6.04e-6 1.53 2.10e-6 0.92 1.11e-6 0.36 8.63e-7 8192 2.14e-5 1.92 5.65e-6 1.73 1.71e-6 1.25 7.19e-7 0.61 4.73e-7 16384 2.12e-5 1.96 5.46e-6 1.85 1.51e-6 1.53 5.24e-7 0.92 2.77e-7 32768 2.11e-5 1.98 5.36e-6 1.92 1.41e-6 1.73 4.27e-7 1.25 1.80e-7 65536 2.11e-5 1.99 5.31e-6 1.96 1.36e-6 1.85 3.78e-7 1.53 1.31e-7 131072 2.11e-5 1.99 5.29e-6 1.98 1.34e-6 1.92 3.53e-7 1.73 1.07e-7 262144 2.11e-5 2.00 5.27e-6 1.99 1.33e-6 1.96 3.41e-7 1.85 9.44e-8Table6 lists the l2-norm error and convergence rates for the fully implicit scheme with various h=2π/Nx and Δt=T/Nt. We have a range of convergence results from the first-order to the second-order accuracy. In the lower left triangular region in Table 6, the magnitude of the spatial discretization error dominates that of the temporal discretization error. Therefore, we have the second-order convergence. In the upper right triangular region in Table 6, the magnitude of the temporal discretization error dominates that of the spatial discretization error. Therefore, we have the first-order convergence. This result implies that if we refine both the time and space steps, then we may have the second-order convergence even though the fully implicit scheme is first-order accurate in temporal discretization.Table 6
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h.
N t \ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5  1.15e-5  7.56e-6  6.58e-6  6.33e-6 rate  1.70  1.37  1.13  1.04 1024 2.42e-5  8.39e-6  4.44e-6  3.45e-6  3.21e-6 rate  1.82  1.54  1.23  1.07 2048 2.26e-5  6.82e-6  2.88e-6  1.89e-6  1.64e-6 rate  1.90  1.70  1.37  1.13 4096 2.18e-5  6.04e-6  2.10e-6  1.11e-6  8.63e-7 rate  1.95  1.82  1.54  1.23 8192 2.14e-5  5.65e-6  1.71e-6  7.19e-7  4.73e-7 rate  1.97  1.90  1.70  1.37 16384 2.12e-5  5.46e-6  1.51e-6  5.24e-7  2.77e-7 rate  1.99  1.95  1.82  1.54 32768 2.11e-5  5.36e-6  1.41e-6  4.27e-7  1.80e-7 rate  1.99  1.97  1.90  1.70 65536 2.11e-5  5.31e-6  1.36e-6  3.78e-7  1.31e-7 rate  2.00  1.99  1.95  1.82 131072 2.11e-5  5.29e-6  1.34e-6  3.53e-7  1.07e-7 rate  2.00  1.99  1.97  1.90 262144 2.11e-5  5.27e-6  1.33e-6  3.41e-7  9.44e-8


## 3.1.1. Convergence Test for the CN Scheme

Table1 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. Here, the temporal convergence rate is defined as log2‖eNxNt‖2/‖eNx2Nt‖2. The boxed rates are the numbers greater than or equal to 1.9. As we refine the space grid size, we have more second-order temporal convergence results.Table 1
l 2-norm error and temporal convergence rates for the CN scheme with various Δt at T=0.1.
N t ∖ N x 100 200 400 800 1600 1 3.23e-5 4.81e-5 5.21e-5 5.31e-5 5.33e-5 rate 2.06 2.58 2.12 2.03 2.01 2 7.73e-6 8.07e-6 1.20e-5 1.30e-5 1.33e-5 rate -1.20 2.06 2.57 2.11 2.03 4 1.77e-5 1.93e-6 2.02e-6 3.00e-6 3.25e-6 rate -0.19 -1.20 2.06 2.57 2.11 8 2.02e-5 4.43e-6 4.82e-7 5.04e-7 7.51e-7 rate -0.04 -0.19 -1.20 2.06 2.57 16 2.08e-5 5.05e-6 1.11e-6 1.21e-7 1.26e-7 rate -0.01 -0.04 -0.19 -1.20 2.06 32 2.10e-5 5.21e-6 1.26e-6 2.77e-7 3.02e-8 rate -0.00 -0.01 -0.04 -0.19 -1.20 64 2.10e-5 5.25e-6 1.30e-6 3.16e-7 6.92e-8 rate -0.00 -0.00 -0.01 -0.04 -0.19 128 2.10e-5 5.26e-6 1.31e-6 3.26e-7 7.90e-8 rate -0.00 -0.00 -0.00 -0.01 -0.04 256 2.10e-5 5.26e-6 1.31e-6 3.28e-7 8.14e-8 rate -0.00 -0.00 -0.00 -0.00 -0.01 512 2.10e-5 5.26e-6 1.32e-6 3.29e-7 8.20e-8Table2 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. The spatial convergence rate is defined as log2‖eNxNt‖2/‖e2NxNt‖2. As we refine the time grid size, we have more second-order spatial convergence results.Table 2
l 2-norm error and spatial convergence rates for the CN scheme with various h at T=0.1.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5 -0.58 4.81e-5 -0.11 5.21e-5 -0.03 5.31e-5 -0.01 5.33e-5 2 7.73e-6 -0.06 8.07e-6 -0.57 1.20e-5 -0.11 1.30e-5 -0.03 1.33e-5 4 1.77e-5 3.20 1.93e-6 -0.06 2.02e-6 -0.57 3.00e-6 -0.11 3.25e-6 8 2.02e-5 2.19 4.43e-6 3.20 4.82e-7 -0.06 5.04e-7 -0.57 7.51e-7 16 2.08e-5 2.04 5.05e-6 2.19 1.11e-6 3.20 1.21e-7 -0.06 1.26e-7 32 2.10e-5 2.01 5.21e-6 2.04 1.26e-6 2.19 2.77e-7 3.20 3.02e-8 64 2.10e-5 2.00 5.25e-6 2.01 1.30e-6 2.04 3.16e-7 2.19 6.92e-8 128 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.01 3.26e-7 2.04 7.90e-8 256 2.10e-5 2.00 5.26e-6 2.00 1.31e-6 2.00 3.28e-7 2.01 8.14e-8 512 2.10e-5 2.00 5.26e-6 2.00 1.32e-6 2.00 3.29e-7 2.00 8.20e-8Table3 lists the l2-norm error and convergence rates for the CN scheme with various h=2π/Nx and Δt=T/Nt. The convergence rate is defined as log2‖eNxNt‖2/‖e2Nx2Nt‖2. We have second-order convergence results for all the cases. We have a sequence of the second-order convergence (diagonal sequence in Table 3) regardless of starting pair, (Nx,Nt). We can find some examples of this approach. In [1], the refinement path was taken to be Δt=0.2h/2 to show the second-order convergence of the proposed numerical scheme for the Cahn–Hilliard–Navier–Stokes equation. In [2], Δt=h was taken to show the second-order convergence for solving space fractional diffusion equations. In [3], Δt=h was used to show the second-order convergence rate for a two-sided space-fractional diffusion equation with variable coefficients.Table 3
l 2-norm error and convergence rates for the CN scheme with various Δt and h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 1 3.23e-5  4.81e-5  5.21e-5  5.31e-5  5.33e-5 rate  2.00  2.00  2.00  2.00 2 7.73e-6  8.07e-6  1.20e-5  1.30e-5  1.33e-5 rate  2.00  2.00  2.00  2.00 4 1.77e-5  1.93e-6  2.02e-6  3.00e-6  3.25e-6 rate  2.00  2.00  2.00  2.00 8 2.02e-5  4.43e-6  4.82e-7  5.04e-7  7.51e-7 rate  2.00  2.00  2.00  2.00 16 2.08e-5  5.05e-6  1.11e-6  1.21e-7  1.26e-7 rate  2.00  2.00  2.00  2.00 32 2.10e-5  5.21e-6  1.26e-6  2.77e-7  3.02e-8 rate  2.00  2.00  2.00  2.00 64 2.10e-5  5.25e-6  1.30e-6  3.16e-7  6.92e-8 rate  2.00  2.00  2.00  2.00 128 2.10e-5  5.26e-6  1.31e-6  3.26e-7  7.90e-8 rate  2.00  2.00  2.00  2.00 256 2.10e-5  5.26e-6  1.31e-6  3.28e-7  8.14e-8 rate  2.00  2.00  2.00  2.00 512 2.10e-5  5.26e-6  1.32e-6  3.29e-7  8.20e-8We have the following relation between errors:(10)log2eNxNt2e2Nx2Nt2=log2eNxNt2eNx2Nt2+log2eNx2Nt2e2Nx2Nt2=log2eNxNt2e2NxNt2+log2e2NxNt2e2Nx2Nt2.From the convergence results in Tables1, 2, and 3, we can confirm this relation. This convergence relation implies that if we refine both the time and space steps, then we may have the second-order convergence even though one of two convergence rates is not second-order accurate. Interestingly, the convergence relation implies that we do not have the second-order convergence results for both the spatial and temporal convergence rates at the same starting pair, (Nx,Nt).


## 3.1.2. Convergence Test for the Fully Implicit Scheme

Table4 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and the fixed space step size h=2π/Nx. As we refine the space grid size, we have more first-order temporal convergence results.Table 4
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt.
N t ∖ N x 100 200 400 800 1600 512 2.73e-5 1.15e-5 7.56e-6 6.58e-6 6.33e-6 rate 0.18 0.46 0.77 0.93 0.98 1024 2.42e-5 8.39e-6 4.44e-6 3.45e-6 3.21e-6 rate 0.10 0.30 0.63 0.87 0.96 2048 2.26e-5 6.82e-6 2.88e-6 1.89e-6 1.64e-6 rate 0.05 0.18 0.46 0.77 0.93 4096 2.18e-5 6.04e-6 2.10e-6 1.11e-6 8.63e-7 rate 0.03 0.10 0.30 0.63 0.87 8192 2.14e-5 5.65e-6 1.71e-6 7.19e-7 4.73e-7 rate 0.01 0.05 0.18 0.46 0.77 16384 2.12e-5 5.46e-6 1.51e-6 5.24e-7 2.77e-7 rate 0.01 0.03 0.10 0.30 0.63 32768 2.11e-5 5.36e-6 1.41e-6 4.27e-7 1.80e-7 rate 0.00 0.01 0.05 0.18 0.46 65536 2.11e-5 5.31e-6 1.36e-6 3.78e-7 1.31e-7 rate 0.00 0.01 0.03 0.10 0.30 131072 2.11e-5 5.29e-6 1.34e-6 3.53e-7 1.07e-7 rate 0.00 0.00 0.01 0.05 0.18 262144 2.11e-5 5.27e-6 1.33e-6 3.41e-7 9.44e-8Table5 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=2π/Nx and the fixed time step size Δt=T/Nt. As we refine the time step size, we have more second-order spatial convergence results.Table 5
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h.
N t ∖ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5 1.25 1.15e-5 0.61 7.56e-6 0.20 6.58e-6 0.06 6.33e-6 1024 2.42e-5 1.53 8.39e-6 0.92 4.44e-6 0.36 3.45e-6 0.11 3.21e-6 2048 2.26e-5 1.73 6.82e-6 1.25 2.88e-6 0.61 1.89e-6 0.20 1.64e-6 4096 2.18e-5 1.85 6.04e-6 1.53 2.10e-6 0.92 1.11e-6 0.36 8.63e-7 8192 2.14e-5 1.92 5.65e-6 1.73 1.71e-6 1.25 7.19e-7 0.61 4.73e-7 16384 2.12e-5 1.96 5.46e-6 1.85 1.51e-6 1.53 5.24e-7 0.92 2.77e-7 32768 2.11e-5 1.98 5.36e-6 1.92 1.41e-6 1.73 4.27e-7 1.25 1.80e-7 65536 2.11e-5 1.99 5.31e-6 1.96 1.36e-6 1.85 3.78e-7 1.53 1.31e-7 131072 2.11e-5 1.99 5.29e-6 1.98 1.34e-6 1.92 3.53e-7 1.73 1.07e-7 262144 2.11e-5 2.00 5.27e-6 1.99 1.33e-6 1.96 3.41e-7 1.85 9.44e-8Table6 lists the l2-norm error and convergence rates for the fully implicit scheme with various h=2π/Nx and Δt=T/Nt. We have a range of convergence results from the first-order to the second-order accuracy. In the lower left triangular region in Table 6, the magnitude of the spatial discretization error dominates that of the temporal discretization error. Therefore, we have the second-order convergence. In the upper right triangular region in Table 6, the magnitude of the temporal discretization error dominates that of the spatial discretization error. Therefore, we have the first-order convergence. This result implies that if we refine both the time and space steps, then we may have the second-order convergence even though the fully implicit scheme is first-order accurate in temporal discretization.Table 6
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h.
N t \ N x 100 rate 200 rate 400 rate 800 rate 1600 512 2.73e-5  1.15e-5  7.56e-6  6.58e-6  6.33e-6 rate  1.70  1.37  1.13  1.04 1024 2.42e-5  8.39e-6  4.44e-6  3.45e-6  3.21e-6 rate  1.82  1.54  1.23  1.07 2048 2.26e-5  6.82e-6  2.88e-6  1.89e-6  1.64e-6 rate  1.90  1.70  1.37  1.13 4096 2.18e-5  6.04e-6  2.10e-6  1.11e-6  8.63e-7 rate  1.95  1.82  1.54  1.23 8192 2.14e-5  5.65e-6  1.71e-6  7.19e-7  4.73e-7 rate  1.97  1.90  1.70  1.37 16384 2.12e-5  5.46e-6  1.51e-6  5.24e-7  2.77e-7 rate  1.99  1.95  1.82  1.54 32768 2.11e-5  5.36e-6  1.41e-6  4.27e-7  1.80e-7 rate  1.99  1.97  1.90  1.70 65536 2.11e-5  5.31e-6  1.36e-6  3.78e-7  1.31e-7 rate  2.00  1.99  1.95  1.82 131072 2.11e-5  5.29e-6  1.34e-6  3.53e-7  1.07e-7 rate  2.00  1.99  1.97  1.90 262144 2.11e-5  5.27e-6  1.33e-6  3.41e-7  9.44e-8


## 3.2. The Allen–Cahn Equation

The initial condition isϕ(x)=0.2cos⁡(2πx) on Ω=(0,1). We use T = 1.0e-5, and ϵ=0.0075. Because there is no analytic solution for (7), we consider a reference solution. We define the reference solution ϕref as the numerical solution with very fine space and time steps, (Δtref, href) = (3.81e-12, 1/2048).


### 3.2.1. Convergence Test for the CN Scheme

Table7 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt with a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the second-order temporal convergence result.Table 7
l 2-norm error and temporal convergence rates for the CN scheme with various Δt. (Δt, h)=(1.25e-7, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 7.13e-9 2.00 1.78e-9 2.00 4.46e-10 2.00 1.11e-10Table8 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=1/Nx with a fixed time step size Δt=T/2621440. As we refine the space grid size, i.e., Nx=8,16,32 and 64, we have the second-order spatial convergence result.Table 8
l 2-norm error and spatial convergence rates for the CN scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8Tables9 and 10 list the l2-norm error and spatial and temporal convergence rates for the CN scheme with various h=1/Nx and Δt=T/Nt. The results generate the second-order convergence for both the cases, i.e., (Δt, h)=(1.25e-7, 1/64) and (Δt, h)=(6.10e-11, 1/8).Table 9
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 6.03e-8 2.01 1.50e-8 2.02 3.69e-9 2.09 8.68e-10Table 10
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8


### 3.2.2. Convergence Test for the Fully Implicit Scheme

Table11 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the first-order temporal convergence result.Table 11
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt. (Δt, h)=(1.25e-07, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 2.82e-5 1.01 1.40e-5 1.01 6.96e-6 1.02 3.42e-6Table12 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=1/2048 with a fixed time step size Δt=T/Nt. As we refine the space grid size, i.e., Nx=8,16,32, and 64, we have the second-order spatial convergence result.Table 12
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.46e-6 1.99 8.69e-7 1.97 2.22e-7 1.86 6.10e-8Tables13 and 14 list the l2-norm error and spatial and temporal convergence rates for the fully implicit scheme with various h=1/Nx and Δt=T/Nt, respectively.Table 13
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 2.85e-5 1.01 1.42e-5 1.00 7.08e-6 1.00 3.54e-6Table 14
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.47e-6 2.00 8.67e-7 2.00 2.17e-7 2.00 5.41e-8Table13 shows that the convergence rate appears to be only first-order with Nt=80 and Nx=64. On the other hand, Table 14 shows that the convergence rate appears to be second-order with Nt=163840 and Nx=8. These results demonstrate that even if the numerical scheme is only first-order in time and second-order in space, we may have the second-order convergence with some refinement combination.


## 3.2.1. Convergence Test for the CN Scheme

Table7 lists the l2-norm error and temporal convergence rates for the CN scheme with various Δt=T/Nt with a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the second-order temporal convergence result.Table 7
l 2-norm error and temporal convergence rates for the CN scheme with various Δt. (Δt, h)=(1.25e-7, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 7.13e-9 2.00 1.78e-9 2.00 4.46e-10 2.00 1.11e-10Table8 lists the l2-norm error and spatial convergence rates for the CN scheme with various h=1/Nx with a fixed time step size Δt=T/2621440. As we refine the space grid size, i.e., Nx=8,16,32 and 64, we have the second-order spatial convergence result.Table 8
l 2-norm error and spatial convergence rates for the CN scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8Tables9 and 10 list the l2-norm error and spatial and temporal convergence rates for the CN scheme with various h=1/Nx and Δt=T/Nt. The results generate the second-order convergence for both the cases, i.e., (Δt, h)=(1.25e-7, 1/64) and (Δt, h)=(6.10e-11, 1/8).Table 9
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 6.03e-8 2.01 1.50e-8 2.02 3.69e-9 2.09 8.68e-10Table 10
l 2-norm error and convergence rates for the CN scheme with various h and Δt. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.45e-6 2.00 8.61e-7 2.01 2.14e-7 2.01 5.32e-8


## 3.2.2. Convergence Test for the Fully Implicit Scheme

Table11 lists the l2-norm error and temporal convergence rates for the fully implicit scheme with various Δt=T/Nt and a fixed space step size h=1/2048. As we refine the time step size, i.e., Nt=80,160,320, and 640, we have the first-order temporal convergence result.Table 11
l 2-norm error and temporal convergence rates for the fully implicit scheme with various Δt. (Δt, h)=(1.25e-07, 1/2048) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) rate (Δt/8, h) l 2-error 2.82e-5 1.01 1.40e-5 1.01 6.96e-6 1.02 3.42e-6Table12 lists the l2-norm error and spatial convergence rates for the fully implicit scheme with various h=1/2048 with a fixed time step size Δt=T/Nt. As we refine the space grid size, i.e., Nx=8,16,32, and 64, we have the second-order spatial convergence result.Table 12
l 2-norm error and spatial convergence rates for the fully implicit scheme with various h. (Δt, h)=(3.81e-12, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) rate (Δt, h/8) l 2-error 3.46e-6 1.99 8.69e-7 1.97 2.22e-7 1.86 6.10e-8Tables13 and 14 list the l2-norm error and spatial and temporal convergence rates for the fully implicit scheme with various h=1/Nx and Δt=T/Nt, respectively.Table 13
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(1.25e-7, 1/64) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 2.85e-5 1.01 1.42e-5 1.00 7.08e-6 1.00 3.54e-6Table 14
l 2-norm error and convergence rates for the fully implicit scheme with various Δt and h. (Δt, h)=(6.10e-11, 1/8) and (Δtref, href) = (3.81e-12, 1/2048) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) rate (Δt/8, h/8) l 2-error 3.47e-6 2.00 8.67e-7 2.00 2.17e-7 2.00 5.41e-8Table13 shows that the convergence rate appears to be only first-order with Nt=80 and Nx=64. On the other hand, Table 14 shows that the convergence rate appears to be second-order with Nt=163840 and Nx=8. These results demonstrate that even if the numerical scheme is only first-order in time and second-order in space, we may have the second-order convergence with some refinement combination.


## 3.3. The Cahn–Hilliard Equation

The initial condition isϕ(x)=0.1cos⁡(2πx) on Ω=(0,1). Table 15 lists the l2-norm error and temporal convergence rates for the unconditionally stable scheme with various Δt=T/Nt with a fixed space step size h=1/Nx. We use Nx=1024, T = 2.40e-3, and ϵ=0.02. We define the reference solution with fine time step Δt = 9.54e-6. We find that the convergence rates are all first-order.Table 15
l 2-norm error and temporal convergence rates for the unconditionally stable numerical scheme with various Δt. (Δt, h)=(3.05e-4, 1/1024) and (Δtref, href</caption>) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h) rate (Δt/4, h) l 2-error 4.71e-5 1.05 2.27e-5 1.11 1.05e-5Table16 lists the l2-norm error and spatial convergence rates with various h=1/Nx with a fixed time step size Δt=T/Nt. We use Δt = 9.54e-6, T = 2.40e-3, and ϵ=0.02. We define the reference solution with fine space step h=1/1024. We can find that the convergence rates are second-order.Table 16
l 2-norm error and spatial convergence rates for the unconditionally stable numerical scheme with various h. (Δt, h)=(9.54e-6, 1/16) and (Δtref, href</caption>) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt, h/2) rate (Δt, h/4) l 2-error 9.90e-5 1.98 2.52e-5 1.96 6.46e-6Table17 lists the l2-norm error and spatial and temporal convergence rates with various h=1/Nx and Δt. We use T = 2.40e-3 and ϵ=0.02. We can find that the convergence rates are first-order.Table 17
l 2-norm error and convergence rates for the unconditionally stable numerical scheme with various Δt and h. (Δt, h)=(1.20e-3, 1/32) and (Δtref, href) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) l 2-error 2.06e-4 1.06 9.87e-5 1.00 4.93e-5Table18 lists the l2-norm error and spatial and temporal convergence rates with various h=1/Nx and Δt. We use T = 2.40e-3 and ϵ=0.02. We can find that the convergence rates are second-order.Table 18
l 2-norm error and convergence rates for the unconditionally stable numerical scheme with various Δt and h. (Δt, h)=(7.63e-5, 1/8) and (Δtref, href) = (9.54e-6, 1/1024) are used.
case (Δt, h) rate (Δt/2, h/2) rate (Δt/4, h/4) l 2-error 3.88e-4 1.91 1.03e-4 1.95 2.67e-5The results in Tables17 and 18 show that we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes in the case of the first-order in time and the second-order in space scheme.


## 4. Conclusion

We presented verification methods for the convergence rates of the numerical solutions for parabolic equations. As examples, we considered the numerical convergence rates of the heat equation, the AC equation, and the CH equation. Convergence test results showed that if we refine the spatial and temporal steps at the same time, then we may have the second-order convergence rate for the fully implicit scheme, which is first-order accurate in time and second-order accurate in space. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.



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