I know there is a solution to this pde

$$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$ $$ f(0,x)=g(x)$$ ( Where $v$ and $g$ are known functions) which is given by $$ f(t,x)=\frac{1}{v(x)} h(t+\int \frac{1}{v(x)})$$ where $h(x)$ is determined by initial condition $g(x)$.

The question is if I use the method of characteristic would it give me the same solution with this initial condition?