A high-order immersed interface method free of derivative jump conditions for Poisson equations on irregular domains

•A functional and easy-to-implement Immersed Interface method for Poisson equations on irregular domains.•The only specific input data required by the present method is the ordinary jumps of the function.•New Finite Difference schemes are constructed so as to avoid the derivative jumps.•High-order approximations are obtained for both 1D and 2D cases.•The resolution method leads to a system of discrete equations which can be solved in a standard fashion.