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I am reading a paper, where the author states that two values $U^{n+1}_{1,m}$ and $U^{n+1}_{2,m}$ can be found by solving:

$$U^{n+1}_{1,m}=U^{n+1}_{1,m-1} + hg_{1}(U^{n+1}_{1,m},U^{n+1}_{2,m})$$ $$U^{n+1}_{2,m}=U^{n+1}_{2,m-1} + hg_{2}(U^{n+1}_{1,m},U^{n+1}_{2,m})$$

Where:

$$g_{1}=K[V_3(U^{n+1}_{1,m})-V_4(U^{n+1}_{1,m})+V_5(U^{n+1}_{1,m},U^{n+1}_{2,m})-V_6(U^{n+1}_{1,m},U^{n+1}_{2,m})]$$

$$g_{2}=K[V_4(U^{n+1}_{1,m})-V_5(U^{n+1}_{1,m}]$$

via Newton's method. I need to be able to replicate their results, I am familiar with Newton's method and I know how to program, but I don't see how to proceed.

How can I find $U^{n+1}_{1,m}$ and $U^{n+1}_{2,m}$ ?

The paper can be found here and the supplementary information containing this claim here

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