APPROXIMATING SOLUTIONS FOR CONSERVATION LAWS USING MODIFIED NUMERICAL METHODS
Alejandra Castillo1, Julio César Enciso Alva2, Crystal Mackey3, Armando Morales4, Cynthia Flores4.
1Pomona College, Claremont, CA, 2Universidad Autónoma del Estado de Hidalgo, Pachuca, Hidalgo, MX, 3Youngstown State University, Youngstown, OH, 4California State University, Channel Islands, Camarillo, CA.
Burger's equation is a non-linear conservation law that is used to model traffic flow, hydrostatics, gas dynamics, and serves as a simplification of the Navier-Stokes equation. Paired with an initial condition, Burger's equation becomes a partial differential equation known as the Burger's equation initial value problem (IVP), which can be solved analytically, though these solutions limit the physical applications of exact solutions. This motivated the use of numerical simulations to approximate exact solutions. Using difference methods, we ran numerical simulations to approximate exact solutions of the Burger's equation IVP. We carefully derived the Backward Euler and Lax-Wendroff difference methods for linear and nonlinear conservation laws. This allowed us to make original modifications to the Lax-Wendroff difference method thus improving our numerical results when comparing them to real data.