This paper presents numerical simulations of liquid-solid and solid-liquid phase change processes using mathematical models in Lagrangian and Eulerian descriptions. The mathematical models are derived by assuming a smooth interface or transition region between the solid and liquid phases in which the specific heat, density, thermal conductivity, and latent heat of fusion are continuous and differentiable functions of temperature. In the derivations of the mathematical models we assume the matter to be homogeneous, isotropic, and incompressible in all phases. The change in volume due to change in density during phase transition is neglected in all mathematical models considered in this paper. This paper describes various approaches of deriving mathematical models that incorporate phase transition physics in various ways, hence results in different mathematical models. In the present work we only consider the following two types of mathematical models: (i) We assume the velocity field to be zero i.e. no flow assumption, and free boundaries i.e. zero stress field in all phases. Under these assumptions the mathematical models reduce to first law of thermodynamics i.e. the energy equation, a nonlinear diffusion equation in temperature if we assume Fourier heat conduction law relating temperature gradient to the heat vector. These mathematical models are invariant of the type of description i.e. Lagrangian or Eulerian due to absence of velocities and stress field. (ii) The second class of mathematica models are derived with the assumption that stress field and velocity field are nonzero in the fluid region but in the solid region stress field is assumed constant and the velocity field is assumed zero. In the transition region the stress field and the velocity field transition in a continuous and differentiable manner from nonzero at the liquid state to constant and zero in the solid state based on temperature in the transition zone. Both of these models are consistent with the principles of continuum mechanics, hence -provide correct interaction between the regions and are shown to work well in the numerical simulations of phase transition applications with flow. Details of other mathematical models, problems associated with them, and their limitations are also discussed in this paper. Numerical solutions of phase transition model problems in R1 and R2 are presented using these two types of mathematical models. Numerical solutions are obtained using h, p, k space-time finite element processes based on residual functional for an increment of time with time marching in which variationally consistent space-time integral forms ensure unconditionally stable computations during the entire evolution.