proved significantly when accelerated by a multipole algorithm.

Multipole expansions result when Laplace's equation is solved in spherical coordinates using the method of separation of variables. The multipole algorithm approximates the influence of groups of far field sources with expansions, thereby replacing the influence coefficient matrix. In a multipole-accelerated iterative solver, the matrix/vector product required at each iteration (normally *O*(*N*^{2}) cost) is replaced by a multipole approximation (*O(N)* cost). Greengard (1987) presents an efficient algorithm for the application of multipole expansions to potential problems. Nabors et al. (1994) presents a slightly different approach developed for electrostatics problems, and Korsmeyer et al. (1993) extend that approach to Laplace problems in general, particularly the Green's theorem formulation of free surface problems.

In the following sections, we present formulations for the boundary value problem for the fluid velocity potential, the desingularized integral equations for the perturbation potential, a domain decomposition solution technique, and the multipole algorithm. These are followed by numerical results. The efficiency of the multipole algorithm is evaluated by solving a fictitious boundary value problem for the Rankine source strength using an iterative solver with and without multipole acceleration. Next, a new method for including fully nonlinear incident waves in the forward speed problem is introduced. Finally, these methods are applied to a submerged submarine moving with constant forward speed under incident waves.

The fluid is assumed inviscid and incompressible. The problem is started from rest so that the flow remains irrotational. This implies that the fluid velocity field can be described by a scalar potential, **Φ**. Consider a vessel floating on a free surface and translating with speed (*t*) with respect to a right handed space fixed coordinate system. The *z*=0 plane defines the calm water level and the *x–z* plane is coincident with the centerplane of the vessel. Since we are interested in ships, we want to concentrate on the “forward speed ” case which is (*t*)=(*U*(*t*),0,0) where *U(t)* is in the negative *x* direction. The total velocity potential describing the fluid motion is then,

(1)

where is the perturbation potential and =(*x, y, z*). Both potentials **Φ** and satisfy Laplace's equation in the fluid domain.

(2)

In order to define a well posed problem, boundary conditions must be specified on all surfaces surrounding the fluid domain. Since the forward speed part of the potential is known, the boundary conditions can be set up in terms of the unknown perturbation potential with respect to the translating coordinate system in the centerplane of the vessel. On the body wetted surface *(S*_{H}*)* and on the bottom *(S*_{B}*)* the boundary conditions are,

(3)

and

(4)

where is the unit normal out of the fluid. *V*_{B} is the velocity of the body surface. On the far field surface at infinity (*S*_{∞}) the boundary condition is,

(5)

Due to the fact that only a finite amount of the free surface can be modeled, this far field condition is usually treated by either an approximate radiation condition, an absorbing beach, walls, or, in some cases, it is ignored and the boundary is left open. If the far field surfaces are truncated with solid walls *(S*_{W}*),* the far field boundary condition is then,

(6)

There are two boundary conditions on the free surface *(S*_{F}*),* one kinematic and one dynamic. We use a special form of these conditions for following the temporal evolution of nodes on the free surface (see Beck et al. 1994). The kinematic condition is,

(7)

(8)