Construction of a physical model for the grand piano implies complex and multidimensional phenomena. We present a model of piano strings coupled to a soundboard, and its numerical approximation. Measurements on piano strings and bridge show phantom partials and a time precursor that both cannot be explained by the linear scalar string model. A classical model of nonlinear strings has been written by Morse & Ingard, it implies to consider the longitudinal displacement as well as the standard transversal displacement of the string, in a nonlinear coupled system. Various approximate (polynomial) models have been written from this one, by expanding the nonlinearity (a square root term) around the rest position of the string. We provide a mathematical justification of the most used model. Transmission of the string motion to the rest of the structure is essential from the acoustical point of view. We use a modal approach for the soundboard, and we write a nonstandard reciprocal coupling condition between strings and soundboard at the bridge. Numerical approximation of such a nonlinear, multidimensional and coupled problem is a difficult issue. We use an energy approach to achieve stability, which leads to an innovating implicit numerical scheme.