## 1 Introduction  
Sound synthesis techniques for string instruments have evolved, in the past few decades, from abstract synthesis [1] (wavetables, frequency-modulation (FM) synthesis...) towards sampling synthesis, based on a library of pre-recorded sounds, and physical models, emulating the instruments themselves. To this day, the best synthesised sound quality is achieved by sampling techniques; however, the potentially very large storage requirements for these sound libraries are a major argument for using physical models[PERSON] Beyond the storage concerns, the use of a physical model allows for great flexibility for input parameters (typically, the instrument's shape and material properties, together with the player's controls), as well as output parameters, usually the \"listening conditions\", that can be changed freely and dynamically along a simulation, as opposed to the case of statically recorded samples[PERSON]  
Physical modelling synthesis for strings debuted in the 1970s, with time stepping methods to discretise and directly solve the 1D wave equation [2; 3; 4]. However, the very limited computational power at the time ruled out simulation at an audio sample rate in any reasonable amount of time. The next generation of models therefore focussed on algorithmic simplification, through physically plausible assumptions. The [PERSON] Karplus-Strong string synthesis algorithm [5; 6] was generalised by the digital waveguide framework [7; 8]; [PERSON] _et al._[9] review the use of these models for string synthesis. Their fast execution and realistic sound output found efficient applications in bowed string modelling, and are still widely used to this day [10; 11; 12; 13; 14]. Digital waveguides model the forward and backward travelling waves along a string using delay lines--a simple and efficient strategy for certain linear time invariant systems. In particular, they are well suited for systems in one dimension, well described by the wave equation. Another more general class of physical models relies on the modal solutions of the string equation, and has been successfully adapted for bowed strings [15; 16]. However, the very assumptions that underlie the efficiency of these methods can lead to difficulties when extensions to more realistic settings are desired--the bowed string and its complex, nonlinear, time-varying interaction with the environment being an excellent example. Time-stepping methods, and more specifically finite difference methods [17], though computationally costly, have regained appeal in musical sound synthesis [18] with the great increase in computing power during the last two decades. String simulation in one dimension is particularly suited for these kind of methods [19; 20]. The interactions of, say, a bowed string with its environment can be included in a straightforward manner, as long as they can be described with a system of partial differential equations (PDEs). These methods also allow a greater flexibility for modelling the musician's gestures, and more generally to deal with the dynamic nature of the input and output parameters. While the number of parameters is small compared to other methods, navigating the space they describe is somewhat of a challenge. Indeed, as opposed to a struck or plucked string, the continuous excitation mechanism of a bowed string makes playability a major issue, for real or virtual instruments. The player shapes the sound and behaviour of his instrument throughout the whole production of a note.