In this section, we develop the 2-SC method described in the previous section to 2D-VIEs of form (1.1).
To this end, let *N* and *M* be positive integers, and consider the uniform grids

${x}_{i}=ik,$$i=0,1,\mathrm{\dots},M,$$Mk\mathit{\hspace{1em}}=X,$${t}_{i}=ih,$$i=0,1,\mathrm{\dots},N,$$Nh\mathit{\hspace{1em}}=T.$

We set $x={x}_{i}$ in (1.1), and thus we have

$y({x}_{i},t)=g({x}_{i},t)+{\int}_{0}^{t}{\int}_{0}^{{x}_{i}}K({x}_{i},t,z,s,y(z,s))dzds.$(3.1)

Now, substituting the inner integral of (3.1) by an appropriate quadrature rule depending on ${x}_{j}$, where $j=0,1,\mathrm{\dots},i$, we obtain

${y}_{i}(t)={g}_{i}(t)+{\int}_{0}^{t}\sum _{j=0}^{i}{w}_{ij}{K}_{ij}(t,s,{y}_{j}(s))ds,$(3.2)

which is a one-dimensional VIE of the second kind, and we solve it by the two-step collocation method described in the previous section.
In equation (3.2), ${y}_{i}(t)$, ${g}_{i}(t)$ and ${K}_{ij}(t,s,\cdot )$ denote $y({x}_{i},t)$, $g({x}_{i},t)$ and $K({x}_{i},t,{x}_{j},s,\cdot )$, respectively, and ${w}_{ij}$ are quadrature weights.
In this procedure, we use the values obtained from the previous steps.

First, setting $x={x}_{1}$ and using the trapezoidal rule, we have

${y}_{1}(t)={g}_{1}(t)+{\int}_{0}^{t}\frac{k}{2}[K({x}_{1},t,0,s,y(0,s))+K({x}_{1},t,{x}_{1},s,{y}_{1}(s))]ds,$

where it is obvious that $y(0,s)=g(0,s)$.

Therefore, applying the two-step collocation method to this equation, we obtain an approximate polynomial to ${y}_{1}(t)=y({x}_{1},t)$, namely, ${P}_{1}(t)$.

For $x={x}_{2}$, we use Simpson’s rule for the interior integral in (3.1).
Thus we obtain

${y}_{2}(t)={g}_{2}(t)+{\int}_{0}^{t}\frac{k}{3}[K({x}_{2},t,0,s,y(0,s))+4K({x}_{2},t,{x}_{1},s,y({x}_{1},s))+K({x}_{2},t,{x}_{2},s,{y}_{2}(s))]ds,$

where $y(0,s)$ and $y({x}_{1},s)$ are known, and therefore we obtain ${P}_{2}(t)$, the approximate polynomial of ${y}_{2}(t)$ using the two-step collocation method.

For $x={x}_{i}$, $i=3,4,\mathrm{\dots},N$, we use Simpson’s rule for even indices and Simpson’s rule with the trapezoidal rule for the last subinterval with odd indices.
From [7], this method (Simpson and trapezoidal) is stable.

To simplify the notation, we set

${K}_{i}(t,s,{y}_{i}(s)):-\sum _{j=0}^{i}{w}_{ij}{K}_{ij}(t,s,{y}_{j}(s)).$

To analyze the error of the presented method, we assume that the maximum error occurs at the *i*th stage, that is, for $x={x}_{i}$.
At the *i*th stage, we have equation (3.1), and substituting the inner integral by a quadrature rule of order, for example, *r*, we have

${y}_{i}(t)={g}_{i}(t)+{\int}_{0}^{t}{K}_{i}(t,\eta ,{y}_{i}(\eta ))d\eta +{A}_{i}{k}^{r}t$(3.3)

with ${\parallel {A}_{i}\parallel}_{\mathrm{\infty}}\le {C}_{1}$ independent of *k*.

The next theorem investigates the error of the presented method.

#### Theorem 3.1.

*Let ${e}_{i\mathrm{,}h}\mathrm{:-}{y}_{i}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{-}{P}_{i}\mathit{}\mathrm{(}t\mathrm{)}$ be the error of the new method at stage **i*.
Suppose that the hypotheses of Theorem 2.1 are satisfied for $p\mathrm{=}\mathrm{2}\mathit{}m\mathrm{-}\mathrm{1}$ for the *i*th stage, with ${\phi}_{\mathrm{0}}\mathit{}\mathrm{(}s\mathrm{)}$ and ${\phi}_{\mathrm{1}}\mathit{}\mathrm{(}s\mathrm{)}$ chosen according to (2.7) and (2.8), respectively.
Moreover, assume that

(i)

$\frac{\partial}{\partial y}{K}_{i}(t,s,\cdot )$
* exists and is bounded for *
$0\le s\le t\le T$,

(ii)

*the quadrature formulas (*
2.3
*) and (*
2.4
*) at the *
*i*
*th stage are of order *
$O({h}^{q})$,

(iii)

*the quadrature formula (*
3.2
*) used for the *
*i*
*th stage is of order *
$O({k}^{r})$,

(iv)

*the starting error is *
${\parallel {e}_{i,h}\parallel}_{\mathrm{\infty},[0,{t}_{1}]}=O({h}^{d})$.

*Then the order of convergence of the method is $O\mathit{}\mathrm{(}{h}^{{p}^{\mathrm{*}}}\mathrm{+}{k}^{r}\mathrm{)}$, where ${p}^{\mathrm{*}}\mathrm{=}\mathrm{min}\mathit{}\mathrm{\{}d\mathrm{+}\mathrm{1}\mathrm{,}q\mathrm{,}\mathrm{2}\mathit{}m\mathrm{\}}$.*

#### Proof.

From the previous section, the approximate polynomial for ${y}_{i}(t)$ at the *i*th stage is

${P}_{i}({t}_{n}+sh)={\phi}_{0}(s){y}_{i,n-1}+{\phi}_{1}(s){y}_{i,n}+\sum _{j=1}^{m}{\chi}_{j}(s){P}_{i}({t}_{n-1,j})+\sum _{j=1}^{m}{\psi}_{j}(s)\left({F}_{i,h}^{[n]}({t}_{n,j})+{\mathrm{\Phi}}_{i,h}^{[n+1]}({t}_{n,j})\right).$(3.4)

Since the functions ${\phi}_{0}(s)$, ${\phi}_{1}(s)$, ${\chi}_{j}(s)$ and ${\psi}_{j}(s)$ satisfy the collocation conditions, setting ${Y}_{i,j}^{[n+1]}:-{P}_{i}({t}_{n,j})$, we have

${Y}_{i,j}^{[n+1]}={F}_{i,h}^{[n]}({t}_{n,j})+{\mathrm{\Phi}}_{i,h}^{[n+1]}({t}_{n,j}),i=0,1,\mathrm{\dots},M,j=0,1,\mathrm{\dots},N.$

Hence polynomial (3.4) is of the form

${P}_{i}({t}_{n}+sh)={\phi}_{0}(s){y}_{i,n-1}+{\phi}_{1}(s){y}_{i,n}+\sum _{j=1}^{m}\left({\chi}_{j}(s){Y}_{i,j}^{[n]}+{\psi}_{j}(s){Y}_{i,j}^{[n+1]}\right).$(3.5)

It follows from Theorem 2.1 and equation (3.3) that

${y}_{i}({t}_{n}+sh)={\phi}_{0}(s){y}_{i}({t}_{n-1})+{\phi}_{1}(s){y}_{i}({t}_{n})+\sum _{j=1}^{m}\left({\chi}_{j}(s){y}_{i}({t}_{n-1,j})+{\psi}_{j}(s){y}_{i}({t}_{n,j})\right)+{h}^{p+1}{R}_{i,m,n}(s)+{k}^{r}{A}_{i,m,n}(s),$(3.6)

with ${\parallel {R}_{i,m,n}\parallel}_{\mathrm{\infty}}\le {C}_{2}$ independent of *h*.
Thus, subtracting (3.6) from (3.5), we obtain

${e}_{i,h}({t}_{n}+sh)={\phi}_{0}(s){e}_{i,n-1}+{\phi}_{1}(s){e}_{i,n}+\sum _{j=1}^{m}\left({\chi}_{j}(s){e}_{i,n,j}+{\psi}_{j}(s){e}_{i,n+1,j}\right)+{h}^{p+1}{R}_{i,m,n}(s)+{k}^{r}{A}_{i,m,n}(s),$(3.7)

where ${e}_{i,n+1,j}={e}_{i,h}({t}_{n,j})$ and ${e}_{i,n}={e}_{i,h}({t}_{n})$.
On the other hand, applying the mean value theorem, hypothesis (i) ensures that

$\begin{array}{cc}& {K}_{i}({t}_{n,j},{t}_{v-1}+sh,{y}_{i}({t}_{v-1}+sh))-{K}_{i}({t}_{n,j},{t}_{v-1}+sh,{P}_{i}({t}_{v-1}+sh))\hfill \\ & =\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{v-1}+sh,{z}_{i,v-1}(s)){e}_{i,h}({t}_{v-1}+sh),v=1,\mathrm{\dots},n+1,\hfill \end{array}$

where ${z}_{i,v-1}(s)$ is between ${y}_{i}({t}_{v-1}+sh)$ and ${P}_{i}({t}_{v-1}+sh)$.

Now by hypothesis (ii) it follows that

${F}_{i,h}^{[n]}({t}_{n,j})+{\mathrm{\Phi}}_{i,h}^{[n+1]}({t}_{n,j})-{g}_{i}({t}_{n,j})-{\int}_{0}^{{t}_{n,j}}{K}_{i}({t}_{n,j},\eta ,{P}_{i}(\eta ))d\eta ={E}_{i,m,n}{h}^{q}$

with ${\parallel {E}_{i,m,n}\parallel}_{\mathrm{\infty}}\le {C}_{3}$ independent of *h*.

Also, from

${y}_{i}({t}_{n,j})-{P}_{i}({t}_{n,j})={F}_{i}^{[n]}({t}_{n,j})+{\mathrm{\Phi}}_{i}^{[n+1]}({t}_{n,j})-{F}_{i,h}^{[n]}({t}_{n,j})-{\mathrm{\Phi}}_{i,h}^{[n+1]}({t}_{n,j}),$

it follows that

$\begin{array}{cc}\hfill {e}_{i,n+1,j}& ={\int}_{0}^{{t}_{n}}{K}_{i}({t}_{n,j},\eta ,{y}_{i}(\eta ))d\eta +{\int}_{{t}_{n}}^{{t}_{n,j}}{K}_{i}({t}_{n,j},\eta ,{y}_{i}(\eta ))d\eta \hfill \\ & -{\int}_{0}^{{t}_{n}}{K}_{i}({t}_{n,j},\eta ,{P}_{i}(\eta ))d\eta -{\int}_{{t}_{n}}^{{t}_{n,j}}{K}_{i}({t}_{n,j},\eta ,{P}_{i}(\eta ))d\eta -{E}_{i,m,n}{h}^{q}\hfill \\ & ={\int}_{0}^{{t}_{n}}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},\eta ,{z}_{i}(\eta )){e}_{i,h}(\eta )d\eta +{\int}_{{t}_{n}}^{{t}_{n,j}}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},\eta ,{z}_{i}(\eta )){e}_{i,h}(\eta )d\eta -{E}_{i,m,n}{h}^{q}\hfill \\ & =h\sum _{v=1}^{n}{\int}_{0}^{1}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{v-1}+sh,{z}_{i}({t}_{v-1}+sh)){e}_{i,h}({t}_{v-1}+sh)ds\hfill \\ & +h{\int}_{0}^{{c}_{j}}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{n}+sh,{z}_{n}(s)){e}_{i,h}({t}_{n}+sh)ds-{E}_{i,m,n}{h}^{q}.\hfill \end{array}$(3.8)

From hypothesis (i),

${e}_{i,h}({t}_{0}+sh)={h}^{d}{V}_{i}(s)$(3.9)

with ${\parallel V\parallel}_{\mathrm{\infty}}\le {C}_{4}$ independent of *h*.

Substituting expressions (3.7) and (3.9) in equation (3.8), we obtain

$\begin{array}{cc}\hfill {e}_{i,n+1,j}& =h{\int}_{0}^{1}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{0}+sh,{z}_{i}({t}_{0}+sh)){h}^{d}{V}_{i}(s)ds\hfill \\ & +h\sum _{v=2}^{n}{\int}_{0}^{1}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{v-1}+sh,{z}_{i}({t}_{v-1}+sh))\hfill \\ & \mathrm{\times}\{{\phi}_{0}(s){e}_{i,v-2}+{\phi}_{1}(s){e}_{i,v-1}+\sum _{j=1}^{m}({\chi}_{j}(s){e}_{i,v-1,j}+{\psi}_{j}(s){e}_{i,v,j})\hfill \\ & +{h}^{p+1}{R}_{i,m,n}(s)+{k}^{r}{A}_{i,m,n}(s)\}ds\hfill \\ & +h{\int}_{0}^{{c}_{j}}\frac{\partial}{\partial y}{K}_{i}({t}_{n,j},{t}_{n}+sh,{z}_{n}(s))\hfill \\ & \mathrm{\times}({\phi}_{0}(s){e}_{i,n-1}+{\phi}_{1}(s){e}_{i,n}+\sum _{j=1}^{m}({\chi}_{j}(s){e}_{i,n,j}+{\psi}_{j}(s){e}_{i,n+1,j})\hfill \\ & +{h}^{p+1}{R}_{i,m,n}(s)+{k}^{r}{A}_{i,m,n}(s))ds+E{}_{i,m,n}h{}^{q}.\hfill \end{array}$

On the other hand, setting $s=1$ in (3.7), we have

${e}_{i,n+1}={\phi}_{0}(1){e}_{i,n-1}+{\phi}_{1}(1){e}_{i,n}+\sum _{j=1}^{m}({\chi}_{j}(1){e}_{i,n,j}+{\psi}_{j}(1){e}_{i,n+1,j})+{h}^{p+1}{R}_{i,m,n}(s)+{k}^{r}{A}_{i,m,n}(s).$(3.10)

Therefore, denoting

${\epsilon}_{i,v}=\left[\begin{array}{c}\hfill {e}_{i,v,1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {e}_{i,v,m}\hfill \end{array}\right],{\mathbf{E}}_{i,n}=\left[\begin{array}{c}\hfill {E}_{i,n,1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {E}_{i,n,m}\hfill \end{array}\right],{\mathbf{A}}_{i,n}=\left[\begin{array}{c}\hfill {A}_{i,n,1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {A}_{i,n,m}\hfill \end{array}\right],$

we obtain

$\begin{array}{cc}& (I-h{B}_{i}^{[n+1]}){\epsilon}_{i,n+1}-h{w}^{[n+1]}{e}_{i,n}\hfill \\ & =h\sum _{v=1}^{n}{B}_{n}^{[v]}{\epsilon}_{i,v}+h\sum _{v=1}^{n}{w}_{n}^{[v]}{e}_{i,v-1}+{h}^{p+2}\sum _{v=2}^{n}{\rho}_{n}^{[v]}+{h}^{p+2}{\rho}^{[n+1]}+{h}^{q}{\mathbf{E}}_{n}+{h}^{d+1}{\mathbf{S}}_{n}+{k}^{r}{\mathbf{A}}_{n},\hfill \end{array}$(3.11)

where the matrices ${B}_{i}^{[n+1]}$, ${B}_{n}^{[v]}$, ${w}^{[n+1]}$, ${w}_{n}^{[v]}$ and the vectors ${\rho}_{n}^{[v]}$, ${\rho}^{[n+1]}$, ${\mathbf{S}}_{n}$ involve the integrals over $[0,{c}_{j}]$ or $[0,1]$ of $\frac{\partial}{\partial y}{K}_{i}$ multiplied by ${\phi}_{0}$, ${\phi}_{1}$, ${\chi}_{j}$, ${\psi}_{j}$, ${R}_{i,m,n}$, ${V}_{i}$ and ${A}_{i,m,n}$.

Put

${\u03f5}_{i,v}=\left[\begin{array}{c}\hfill {\epsilon}_{i,v+1}\hfill \\ \hfill {e}_{i,v}\hfill \end{array}\right].$

Then, from (3.10) and (3.11), it follows that

$\parallel {\u03f5}_{i,n+1}\parallel \le h{D}_{1}\sum _{v=1}^{n}\parallel {\u03f5}_{i,v}\parallel +{D}_{2}\sum _{v=n-2}^{n}\parallel {\u03f5}_{i,v}\parallel +{\gamma}_{1}{h}^{d+1}+{\gamma}_{2}{h}^{q}+{\gamma}_{3}{h}^{2m}+{\gamma}_{4}{k}^{r},$

where ${D}_{1},{D}_{2},{\gamma}_{1},{\gamma}_{2},{\gamma}_{3},{\gamma}_{4}$ are upper bounds of the norm of vectors and matrices appearing in (3.11).

Hence, using the Gronwall inequality, it follows that

$\parallel {\u03f5}_{i,n+1}\parallel \le C\gamma ({h}^{{p}^{*}}+{k}^{r})=O({h}^{{p}^{*}}+{k}^{r}),$

where $\gamma =\mathrm{max}\{{\gamma}_{1},\mathrm{\dots},{\gamma}_{4}\}$ and *C* is a constant independent of *h* and *k*.
∎

To analyze the stability of method, we define the following notations for the *i*th stage of the method:

$\begin{array}{c}\hfill {b}_{i}=\left[\begin{array}{c}\hfill {b}_{i,1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {b}_{i,m}\hfill \end{array}\right],{w}_{i,0}=\left[\begin{array}{c}\hfill {w}_{i,1,0}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {w}_{i,m,0}\hfill \end{array}\right],{w}_{i,m+1}=\left[\begin{array}{c}\hfill {w}_{i,1,m+1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {w}_{i,m,m+1}\hfill \end{array}\right],\hfill \\ \hfill {W}_{i}={({w}_{i,j,l})}_{j,l=1}^{m},e=[1,\mathrm{\dots},1],A={[{\chi}_{j}({c}_{i})]}_{i,j=1}^{m},B={[{\psi}_{j}({c}_{i})]}_{i,j=1}^{m},\hfill \end{array}$

where ${b}_{i,j}$s and ${w}_{i,j,l}$s are the weights of quadrature rules of the *i*th stage for ${F}_{i,h}^{[n]}$ and ${\mathrm{\Phi}}_{i,h}^{[n+1]}$, respectively.

The following theorem can be obtained analogously to the theorem of [4] for each stage of the presented method.

#### Theorem 3.2.

*Applying the presented method to test equation (2.10) for each stage of the method, leads to the matrix recurrence relation*

$\left[\begin{array}{c}\hfill {y}_{i,n+1}\hfill \\ \hfill {Y}_{i}^{[n+1]}\hfill \\ \hfill {F}_{i,h}^{[n]}\hfill \\ \hfill {y}_{i,n}\hfill \end{array}\right]={M}_{i}(z)\left[\begin{array}{c}\hfill {y}_{i,n}\hfill \\ \hfill {Y}_{i}^{[n]}\hfill \\ \hfill {F}_{i,h}^{[n-1]}\hfill \\ \hfill {y}_{i,n-1}\hfill \end{array}\right],$

*where $z\mathrm{=}h\mathit{}\lambda $ and the stability matrix ${M}_{i}\mathit{}\mathrm{(}z\mathrm{)}$ is
${M}_{i}\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}{P}_{i}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}z\mathrm{)}\mathit{}{Q}_{i}\mathit{}\mathrm{(}z\mathrm{)}$
with*

${P}_{i}(z)=\left[\begin{array}{cccc}\hfill -zB{w}_{i,m+1}\hfill & \hfill I-zB{W}_{i}\hfill & \hfill -B\hfill & \hfill 0\hfill \\ \hfill 1-z{\psi}^{T}(1){w}_{i,m+1}\hfill & \hfill -z{\psi}^{T}(1){W}_{i}\hfill & \hfill -{\psi}^{T}(1)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right],$${Q}_{i}(z)=\left[\begin{array}{cccc}\hfill {\phi}_{1}(c)+zB{w}_{i,0}\hfill & \hfill A\hfill & \hfill 0\hfill & \hfill {\phi}_{0}(c)\hfill \\ \hfill {\phi}_{1}(1)+z{\psi}^{T}(1){w}_{i,0}\hfill & \hfill {\chi}^{T}(1)\hfill & \hfill 0\hfill & \hfill {\phi}_{0}(1)\hfill \\ \hfill z{b}_{i,m+1}\text{\mathit{e}}\hfill & \hfill z\text{\mathit{e}}{b}_{i}^{T}\hfill & \hfill 1\hfill & \hfill z{b}_{i,0}\text{\mathit{e}}\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right].$

Therefore, defining

${V}_{i,n}=\left[\begin{array}{c}\hfill {y}_{i,n}\hfill \\ \hfill {Y}_{i}^{[n]}\hfill \\ \hfill {F}_{i,h}^{[n-1]}\hfill \\ \hfill {y}_{i,n-1}\hfill \end{array}\right]\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{V}_{n}={[{V}_{1,n},\mathrm{\dots},{V}_{N,n}]}^{T},$

we obtain the matrix recurrence relation of the method in the general case as
${V}_{n+1}=M(z){V}_{n}$,
where the stability matrix $M(z)$ is
$M(z)=\mathrm{diag}({M}_{1}(z),\mathrm{\dots},{M}_{N}(z))$.

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