<p>Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.</p>

La méthode IIM (Immersed Interface Method) permet d'étendre certaines méthodes numériques à des problèmes présentant des discontinuités. Elle est utilisée ici pour étudier un fluide incompressible régi par les équations de Navier-Stokes, dans lequel est immergée une membrane exerçant une force singulière. Nous utilisons une méthode de projection dans une grille de différences finies de type MAC. Une dérivation très complète des conditions de saut dans le cas où la viscosité est continue est présentée en annexe. Deux exemples numériques sont présentés : l'un sans membrane, et l'un où la membrane est immobile. Le cas général d'une membrane mobile est aussi étudié en profondeur. / The Immersed Interface Method allows us to extend the scope of some numerical methods to discontinuous problems. Here we use it in the case of an incompressible fluid governed by the Navier-Stokes equations, in which a membrane is immersed, inducing a singular force. We use a projection method and staggered (MAC-type) finite difference approximations. A very complete derivation for the jump conditions is presented in the Appendix, for the case where the viscosity is continuous. Two numerical examples are shown : one without a membrane, and the other where the membrane is motionless. The general case of a moving membrane is also thoroughly studied.

Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.

La méthode IIM (Immersed Interface Method) permet d'étendre certaines méthodes numériques à des problèmes présentant des discontinuités. Elle est utilisée ici pour étudier un fluide incompressible régi par les équations de Navier-Stokes, dans lequel est immergée une membrane exerçant une force singulière. Nous utilisons une méthode de projection dans une grille de différences finies de type MAC. Une dérivation très complète des conditions de saut dans le cas où la viscosité est continue est présentée en annexe. Deux exemples numériques sont présentés : l'un sans membrane, et l'un où la membrane est immobile. Le cas général d'une membrane mobile est aussi étudié en profondeur. / The Immersed Interface Method allows us to extend the scope of some numerical methods to discontinuous problems. Here we use it in the case of an incompressible fluid governed by the Navier-Stokes equations, in which a membrane is immersed, inducing a singular force. We use a projection method and staggered (MAC-type) finite difference approximations. A very complete derivation for the jump conditions is presented in the Appendix, for the case where the viscosity is continuous. Two numerical examples are shown : one without a membrane, and the other where the membrane is motionless. The general case of a moving membrane is also thoroughly studied.