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The aim of this paper is to study jumping numbers and multiplier ideals ofany ideal in a two-dimensional local ring with a rational singularity. Inparticular we reveal which information encoded in a multiplier ideal determinesthe next jumping number. This leads to an algorithm to compute sequentially thejumping numbers and the whole chain of multiplier ideals in any desired range.As a consequence of our method we develop the notion of jumping divisor thatallows to describe the jump between two consecutive multiplier ideals. Inparticular we find a unique minimal jumping divisor that is studiedextensively.

Multiplier ideals in two-dimensional local rings with rational singularities