Correction to: Well-Balanced Fifth-Order Finite Volume WENO Schemes with Constant Subtraction Technique for Shallow Water Equations

Correction to: Journal of Scientific Computing (2025) 102:32 https://doi.org/10.1007/s10915-024-02756-2

In this article, Equations (1.1), (2.1), (2.2), (2.3), (2.6), (2.13), (3.1), (3.2), (3.7) and few in line equations were processed incorrectly.

It should have been appeared as given below.

• 1. Equation (1.1)

  Incorrect Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix} {[}c] h\\ hu \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix} {[}c] hu\\ hu^2+\frac{1}{2}gh^2 \end{bmatrix} = \begin{bmatrix} {[}c] 0\\ -ghb_x \end{bmatrix}, \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix} h\\ hu \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix} hu\\ hu^2+\frac{1}{2} gh^2 \end{bmatrix} = \begin{bmatrix} 0\\ -ghb_x \end{bmatrix}. \end{aligned}$$

• 2. Equation (2.1)

  Incorrect Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \underbrace{\begin{bmatrix} {[}c] H\\ hu \end{bmatrix}}_{{\textbf{U}}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix} {[}c] hu\\ \frac{(hu)^2}{ H-b}+\frac{1}{2}gH^2-gHb \end{bmatrix}}_{\mathcal {F}({\textbf{U}})} = \underbrace{\begin{bmatrix} {[}c] 0\\ -gHb_x \end{bmatrix}}_{\mathcal {S}(H, b_{x})}. \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \underbrace{\begin{bmatrix} H\\ hu \end{bmatrix}}_{\textbf{U}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix} hu\\ \frac{(hu)^2}{ H-b}+\frac{1}{2} gH^2-gHb \end{bmatrix}}_{\mathcal {F}(\textbf{U})} = \underbrace{\begin{bmatrix} 0\\ -gHb_x \end{bmatrix}}_{\mathcal {S}(H, b_{x})}. \end{aligned}$$

• 3. Equation (2.2)

  Incorrect Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \underbrace{\begin{bmatrix}{[}c] H\\ hu \end{bmatrix}}_{{\textbf{U}}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix}{[}c] hu\\ \frac{(hu)^2}{ H-b}+g({\overline{H}}-H)b+\frac{1}{2}gH^2 \end{bmatrix}}_{{\textbf{F}}({\textbf{U}})} = \underbrace{\begin{bmatrix}{[}c] 0\\ g({\overline{H}}-H)b_x \end{bmatrix}}_{{\textbf{S}}(H, b_{x})}, \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \underbrace{\begin{bmatrix} H\\ hu \end{bmatrix}}_{\textbf{U}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix} hu\\ \frac{(hu)^2}{ H-b}+g(\overline{H}-H)b+\frac{1}{2} gH^2 \end{bmatrix}}_{\textbf{F}(\textbf{U})} = \underbrace{\begin{bmatrix} 0\\ g(\overline{H}-H)b_x \end{bmatrix}}_{\textbf{S}(H, b_{x})}. \end{aligned}$$

• 4. Equation (2.3)

  Incorrect Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix}{[}c] H\\ hu \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix}{[}c] hu\\ \mathcal {M}(u) \end{bmatrix} = \begin{bmatrix}{[}c] 0\\ g({\overline{H}}-H)b_x \end{bmatrix}. \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix} H\\ hu \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix} hu\\ \mathcal {M}(u) \end{bmatrix} = \begin{bmatrix} 0\\ g(\overline{H}-H)b_x \end{bmatrix}. \end{aligned}$$

• 5. Equation (2.6)

  Incorrect Equation:

  $$\begin{aligned} {\hat{{\textbf{F}}}}_{i+\frac{1}{2}} = \frac{1}{2}\left( \begin{bmatrix}{[}c] (hu)^-_{i+\frac{1}{2}} \\ \mathcal {M}(u)^-_{i+\frac{1}{2}} \end{bmatrix} +\begin{bmatrix}{[}c] (hu)^+_{i+\frac{1}{2}} \\ \mathcal {M}(u)^+_{i+\frac{1}{2}} \end{bmatrix}\right) -\frac{\alpha }{2}\left( \begin{bmatrix}{[}c] h^+_{i+\frac{1}{2}} \\ (hu)^+_{i+\frac{1}{2}} \end{bmatrix} -\begin{bmatrix}{[}c] h^-_{i+\frac{1}{2}} \\ (hu)^-_{i+\frac{1}{2}} \end{bmatrix}\right) , \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \hat{\textbf{F}}_{i+\frac{1}{2}} = \frac{1}{2}\left( \begin{bmatrix} (hu)^-_{i+\frac{1}{2}} \\ \mathcal {M}(u)^-_{i+\frac{1}{2}} \end{bmatrix} +\begin{bmatrix} (hu)^+_{i+\frac{1}{2}} \\ \mathcal {M}(u)^+_{i+\frac{1}{2}} \end{bmatrix}\right) -\frac{\alpha }{2}\left( \begin{bmatrix} h^+_{i+\frac{1}{2}} \\ (hu)^+_{i+\frac{1}{2}} \end{bmatrix} -\begin{bmatrix} h^-_{i+\frac{1}{2}} \\ (hu)^-_{i+\frac{1}{2}} \end{bmatrix}\right) . \end{aligned}$$

• 6. The equation in the sentence starts with “To compute the source term......” should have been corrected as given below

  “ $\hat{\textbf{F}}_{i+\frac{1}{2}}-\hat{\textbf{F}}_{i-\frac{1}{2}}$” $\Rightarrow$ “$\hat{ \textbf{F}}^{*}_{i+\frac{1}{2}}-\hat{\textbf{F}}^{*}_{i-\frac{1}{2}}$”.

• 7. In Proposition 2.1, the in line equation should be corrected

  Incorrect equation:

  $$\int _{I_i}{\textbf{S}}(H,b_x)dx = \begin{bmatrix}{[}c] 0 \\ \int _{I_i}g({\overline{H}}-H)b_x dx \end{bmatrix} =\begin{bmatrix}{[}c] 0 \\ 0 \end{bmatrix}.$$

  Corrected Equation:

  $$\int _{I_i}\textbf{S}(H,b_x)dx = \begin{bmatrix} 0 \\ \int _{I_i}g(\overline{H}-H)b_x dx \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \end{bmatrix}.$$

• 8. Equation (2.13)

  Incorrect Equation:

  $$\begin{aligned}&\overline{h}_{i}^{n+1}=\overline{h}_{i}^{n}-\lambda \left( \hat{\textbf{F}}^{*,H}_{i+\frac{1}{2}} -\hat{\textbf{F}}^{*,\,h}_{i-\frac{1}{2}}\right) . \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned}&\overline{h}_{i}^{n+1}=\overline{h}_{i}^{n}-\lambda \left( \hat{\textbf{F}}^{*,H}_{i+\frac{1}{2}} -\hat{\textbf{F}}^{*,H}_{i-\frac{1}{2}}\right) . \end{aligned}$$

  .

• 9. Equation (3.1)

  Incorrect Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix} h\\ hu\\ hv\\ \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix}{[}c] h u\\ h u^{2}+\frac{1}{2} g h^{2}\\ h u v\\ \end{bmatrix} + \frac{\partial }{\partial y} \begin{bmatrix}{[}c] h v\\ h u v \\ h v^{2}+\frac{1}{2} g h^{2}\\ \end{bmatrix} = \begin{bmatrix}{[}c] 0\\ -g h b_{x}\\ -g h b_{y}\\ \end{bmatrix}. \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \begin{bmatrix} h\\ hu\\ hv\\ \end{bmatrix} +\frac{\partial }{\partial x} \begin{bmatrix} h u\\ h u^{2}+\frac{1}{2} g h^{2}\\ h u v\\ \end{bmatrix} + \frac{\partial }{\partial y} \begin{bmatrix} h v\\ h u v \\ h v^{2}+\frac{1}{2} g h^{2}\\ \end{bmatrix} = \begin{bmatrix} 0\\ -g h b_{x}\\ -g h b_{y}\\ \end{bmatrix}. \end{aligned}$$

• 10. Equation (3.2)

  Incorrect Equation:

  $$\begin{aligned}&\frac{\partial }{\partial t} \underbrace{\begin{bmatrix}{[}c] H\\ hu\\ hv \end{bmatrix}}_{{\textbf{U}}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix}{[}c] hu\\ \mathcal {M}(u)\\ \frac{(h u)(h v)}{H-b} \end{bmatrix}}_{{\textbf{F}}({\textbf{U}})} +\frac{\partial }{\partial y} \underbrace{\begin{bmatrix}{[}c] hv\\ \frac{(h u)(h v)}{H-b}\\ \mathcal {M}(v) \end{bmatrix}}_{{\textbf{G}}({\textbf{U}})}= \underbrace{\begin{bmatrix}{[}c] 0 \\ S_1(H, b_x)\\ S_2(H, b_y) \end{bmatrix}}_{{\textbf{S}}(H, b)}, \end{aligned}$$

  Corrected Equation:

  $$\begin{aligned} \frac{\partial }{\partial t} \underbrace{\begin{bmatrix} H\\ hu\\ hv \end{bmatrix}}_{\textbf{U}} +\frac{\partial }{\partial x} \underbrace{\begin{bmatrix} hu\\ \mathcal {M}(u)\\ \frac{(h u)(h v)}{H-b} \end{bmatrix}}_{\textbf{F}(\textbf{U})} +\frac{\partial }{\partial y} \underbrace{\begin{bmatrix} hv\\ \frac{(h u)(h v)}{H-b}\\ \mathcal {M}(v) \end{bmatrix}}_{\textbf{G}(\textbf{U})}= \underbrace{\begin{bmatrix} 0 \\ S_1(H, b_x)\\ S_2(H, b_y) \end{bmatrix}}_{\textbf{S}(H, b)}. \end{aligned}$$

• 11. Equation (3.7) should be corrected as given below:

  “ $\mathcal {L}(\overline{\textbf{U}}_{i,j})$” $\Rightarrow$ “$\mathcal {L}_{i,j}(\overline{\textbf{U}})$”.

• 12. In Proposition 3.1, the in line equation should be corrected as given below

  Incorrect equation:

  $$\int _{I_{ij}}{\textbf{S}}(H,b)dxdy = \begin{bmatrix}{[}c] 0 \\ \int _{I_{ij}}g({\overline{H}}-H)b_x dxdy\\ \int _{I_{ij}}g({\overline{H}}-H)b_y dxdy \end{bmatrix} =\begin{bmatrix}{[}c] 0 \\ 0 \\ 0 \end{bmatrix}.$$

  Corrected Equation:

  $$\int _{I_{i,j}}\textbf{S}(H,b)dxdy = \begin{bmatrix} 0 \\ \int _{I_{i,j}}g(\overline{H}-H)b_x dxdy\\ \int _{I_{i,j}}g(\overline{H}-H)b_y dxdy \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.$$

• 13. In Appendix B, Lemma 5.1, the in line equation should be corrected as given below

  “ $\hat{\textbf{F}}^{h}$” $\Rightarrow$ “$\hat{\textbf{F}}^{H}$”.

The original article has been corrected.