| The first part of this thesis is concerned with the devising of a numerical method for the problem of propagating phase transitions in solids, e.g., elastic bars, which is modeled by a 2 x 2 system of conservation laws with an elliptic region. An interesting feature of the equations modeling this phenomenon is that the entropy condition is not enough to identify the physically relevant solution. Therefore, a so-called kinetic relation has to be given. For a special class of elastic materials, it is known that adding viscosity and capillarity to the elastic part of the stress is equivalent to the imposition of a particular kinetic relation. The solutions obtained in this way are called the viscosity-capillarity solutions.;The main objective of this first part is to devise a simple finite difference scheme that produces approximations to the viscosity-capillarity solutions of the equations that govern the propagation of phase transitions in solids (or to the equations of van der Waals fluids) for all positive values of the dimensionless parameter that characterizes the viscosity-capillarity solution. Numerical experiments displaying the convergence properties of the method are presented.;The objective of the second part of this thesis is to obtain a simple a posteriori error estimate for numerical methods for nonlinear scalar conservation laws. The estimate is totally independent (i) of the dimension of the space, (ii) of the type of nonlinearity f, and (iii) of the numerical method. Thus, it can thus be used to define mathematically sound adaptivity algorithms for conservation laws regardless of the numerical schemes used to compute the approximate solution. Preliminary numerical results displaying the sharpness of the estimate are presented. |
