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Let $R=\mathbb{Z}_{4}[v]/\langlev^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an oddpositive integer. Then $R$ is a local non-principal ideal ring of $16$ elementsand there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto$\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality. First, acanonical form decomposition and the structure for any negacyclic code over $R$of length $2n$ are presented. From this decomposition, a completeclassification of all these codes is obtained. Then the cardinality and thedual code for each of these codes are given, and self-dual negacyclic codesover $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all$3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them arepresented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ newand good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ withbasic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basicparameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Grayimages of self-dual negacyclic codes over $R$ of length $14$ are listed.

Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images