A new method is presented for numerically capturing a moving interface of arbitrary dimension and codimension. The method is named the ' local level set method ', since it localizes the level set method near the interface to significantly reduce the computational expense of the level set method. Following the framework of the level set method, an interface is implicitly represented as the zero level set of a vector valued function. A spatial tree structure is used to locally sample the vector valued function near the interface. Using a Lipschitz stable interpolation and a semi-Lagrangian scheme, our method is stable under both the maximum norm and the Lipschitz semi-norm. Due to this stability, the method does not need to reinitialize a level set function. Several numerical examples with high codimension are successfully tested.