Friction force is cumbersome in numerical simulations due to its discontinuity at zero velocity. Fixed-step simulation techniques are especially desirable for control purposes, such as haptic friction rendering and friction compensation. Previous techniques have difficulties especially in numerical robustness and extensibility to multidimensional cases. This paper proposes two discrete-time friction models that can be used in fixed-step simulations. They can be used in multidimensional space, and can capture arbitrary velocity-dependent friction phenomena. The first one is a discrete-time discontinuous model. This model determines the friction force so that the velocity reaches zero in finite time by using the values of the mass and the timestep size. The second one is a discrete-time continuous model, which is a serial coupling of the discontinuous model and a linear viscoelastic element. This model is useful for haptic rendering because it is formulated as a velocity-input, force-output system. The second model can be extended into a more sophisticated friction model, which exhibits a hysteresis behavior in the presliding regime. Results of numerical simulations and an experiment are presented.