We propose a family of methods for simulating two-dimensional incompressible, low Reynolds number flow around a moving obstacle whose motion is prescribed. The methods make use of a universal mesh: a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle's boundary. The resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off-the-shelf time integrators. We demonstrate the approach by using Taylor-Hood elements to approximate the fluid velocity and pressure. To integrate in time, we consider implicit Runge-Kutta schemes as well as a fractional step scheme. We illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations.Organization. This paper is organized as follows. In Section 2, we recall the governing equations for incompressible, viscous flow around a moving obstacle with prescribed evolution, and we recast the equations in weak form. In Section 3, we propose a discretization of the aforementioned equations using a universal mesh in conjunction with Taylor-Hood finite elements [61]. To integrate in time, we propose the use of implicit Runge-Kutta schemes as well as a fractional step scheme. In Section 4, we apply the proposed methods to simulate flow around various obstacles with prescribed evolution: The setup we have described thus far offers the freedom to employ a time integrator of one's choosing to numerically integrate (17), a system of DAEs of index 2, from t D t n 1 to t D t n . We present two examples of integration schemes: a singly diagonally implicit Runge-Kutta (SDIRK) scheme [69,70] and a fractional step scheme [71][72][73]. In accordance with common guidelines for numerically solving DAEs, the SDIRK schemes we consider are stiffly accurate (and hence L-stable) methods [70,74]. The same schemes are considered by, for instance, [75,76] in their studies of high-order methods for the Navier-Stokes equations on fixed domains.We immersed the oscillating disk in a domain D D OE 6; 6 OE 3; 3 and prescribed boundary conditions ‡ In the case of the fractional step scheme, the error in p was measured at t D T t=2 rather than at t D T .