Numerical schemes for nonlinear partial differential equations in semiconductor modeling and calculus of variations

In this dissertation we study two problems: calculating solutions of constrained optimization problems and schemes for equations of Electro-Hydrodynamic. The two problems seem unrelated, but the common ground concerns the delicate numerical methods needed to solve the nonlinear evolutionary partial differential equations that support discontinuous or steep gradient solutions. In the first part we solve the resulting Euler-Lagrange equations via introduction of time as an auxiliary variable. We define the notion of weak solutions of the evolution equation and identify the proper numerical schemes. In the second part we apply ENO schemes to solve the equations of the Electro-Hydrodynamics. We present the numerical results for simulation of a silicon diode. We present numerical simulations of shock waves in a device. In addition we present a new way of splitting source terms from convection with arbitrarily high order of accuracy.