# A Compact Numerical Scheme for the Heat Transfer of Mixed Convection Flow in Quantum Calculus
## 4 Convergence of Numerical Scheme
** Let the first and second stage of the corresponding exact scheme for Equation (37) be, respectively, expressed as  
\[\frac{\overline{\mathbf{u}}_{i+1}^{n+1}-\mathbf{u}_{i+1}^{n}}{\Delta t(b+2)}+\frac{b( \mathbf{u}_{i}^{n+1}-\mathbf{u}_{i}^{n})}{\Delta t}+\frac{\mathbf{u}_{i-1}^{n+1}-\mathbf{u}_{ i-1}^{n}}{\Delta t}=\frac{A}{4}(1+q)\delta_{y}^{2}\mathbf{u}_{i}^{n}+\frac{B}{2(b+2)} \big{(}\mathbf{u}_{i+1}^{n}+b\mathbf{u}_{i}^{n}+\mathbf{u}_{i-1}^{n}\big{)} \tag{42}\]\[\frac{\mathbf{u}_{i+1}^{n+1}-\mathbf{u}_{i+1}^{n}}{\Delta t(b+2)}+\frac{b(\mathbf{u}_{i}^{n+1} -\mathbf{u}_{i}^{n})}{\Delta t(b+2)}+\frac{\mathbf{u}_{i-1}^{n+1}-\mathbf{u}_{i-1}^{n}}{ \Delta t}=\frac{A}{2}(1+q)\delta_{y}^{2}\overline{\mathbf{u}}_{i}^{n+1}+\frac{B}{( b+2)}\Big{(}\overline{\mathbf{u}}_{i+1}^{n+1}+b\overline{\mathbf{u}}_{i}^{n+1}+ \overline{\mathbf{u}}_{i-1}^{n+1}\Big{)} \tag{43}\]  
Let the error between the proposed and exact scheme be expressed as  
\[\overline{\mathbf{u}}_{i}^{n+1}-\overline{\mathbf{U}}_{i}^{n+1}=\overline{\mathbf{e}}_{i} ^{n+1},\ \mathbf{u}_{i}^{n}-\mathbf{U}_{i}^{n}=\mathbf{e}_{i}^{n},\ \mathbf{u}_{i}^{n+1}-\mathbf{U}_{i}^{n+1}=\mathbf{e}_{i}^{n+1} \tag{44}\]  
Subtracting Equation (40) from Equation (42) results in  
\[\frac{\overline{\mathbf{e}}_{i+1}^{n+1}-\mathbf{e}_{i+1}^{n}}{\Delta t(b+2)}+\frac{b( \overline{\mathbf{e}}_{i}^{n+1}-\mathbf{e}_{i}^{n})}{\Delta t(b+2)}+\frac{\overline{ \mathbf{e}}_{i-1}^{n+1}-\mathbf{e}_{i-1}^{n}}{\Delta t(b+2)}=\frac{A(1+q)}{4(\Delta x )^{2}}\left( \begin{array}{c}\mathbf{e}_{i+1}^{n}-2\mathbf{e}_{i}^{n}\\ +\mathbf{e}_{i-1}^{n}\end{array} \right)+\frac{B}{2(b+2)}\left( \begin{array}{c} \mathbf{e}_{i+1}^{n}+b\mathbf{e}_{i}^{n}\\ +\mathbf{e}_{i-1}^{n}\end{array} \right) \tag{45}\]  
Re-write Equation (45) as  
\[\overline{\mathbf{e}}_{i}^{n+1}=\mathbf{e}_{i}^{n}+\frac{1}{b}\Big{[}-\overline{\mathbf{e }}_{i+1}^{n+1}+\mathbf{e}_{i+1}^{n}-\overline{\mathbf{e}}_{i-1}^{n+1}+\mathbf{e}_{i-1}^{n} \Big{]}+\left\{ \begin{array}{c}\frac{\Delta t(1+q)(b+2)}{4(\Delta x)^{2}} \big{(}\mathbf{e}_{i}^{n}-2\mathbf{e}_{i}^{n}+\mathbf{e}_{i-1}^{n}\big{)}+\\ \frac{B\Delta t}{2}\big{(}\mathbf{e}_{i+1}^{n}+b\mathbf{e}_{i}^{n}+\mathbf{e}_{i-1}^{n} \big{)}\end{array} \right\} \tag{46}\]  
Applying \(\|.\|_{\infty}\) on both sides of Equation (46) and letting  
\[ \begin{array}{c}\overline{\mathbf{e}}^{n+1}=\text{max}\bigg{(}\max_{1\leq i\leq N }\Big{|}\overline{\mathbf{e}}_{1,i}^{n+1}\Big{|},\ \max_{1\leq i\leq N}\Big{|}\overline{\mathbf{e}}_{2,i}^{n+1}\Big{|}\bigg{)},\ \text{where}\ \overline{\mathbf{e}}_{i}^{n+1}=\Big{[} \overline{\mathbf{e}}_{1,i}^{n+1},\ \overline{\mathbf{e}}_{2,i}^{n+1}\Big{]}^{t}\\ \overline{\mathbf{e}}^{n+1}\leq e^{n}+\frac{1}{|b|}\Big{(}\overline{\mathbf{e}}^{n+1}+ \mathbf{e}^{n}+\overline{\mathbf{e}}^{n+1}+\mathbf{e}^{n}\Big{)}+\frac{\|A\|_{\infty} \Delta t(1+q)(b+2)}{(\Delta x)^{2}}(\mathbf{e}^{n})+\frac{\|B\|_{\infty}\Delta t( |b|+2)}{2|b|}(\mathbf{e}^{n})\end{array}  \tag{47}\]  
Inequality (47) can be expressed as  
\[\overline{\mathbf{e}}^{n+1}\leq\frac{|b|+2}{|b|-2}e^{n}+\frac{\|\ A\ \|_{\infty}\Delta t(1+q)(b+2)|b|}{(\Delta x)^{2}(|b|-2)}(\mathbf{e}^{n})+\frac{ \Delta t|b|(|b|+2)\|\ B\ \|_{\infty}}{2(|b|-2)}(\mathbf{e}^{n}) \tag{48}\]  
Re-writing inequality (48) gives  
\[\overline{\mathbf{e}}^{n+1}\leq\delta_{1}\mathbf{e}^{n} \tag{49}\]  
where \(\delta_{1}=\frac{|b|+2}{|b|-2}+\frac{\|A\|_{\infty}\Delta t(1+q)(b+2)|b|}{( \Delta x)^{2}(|b|-2)}+\frac{\Delta t|b|(|b|+2)\|\ B\|_{\infty}}{2(|b|-2)}\).