Abstract
In this thesis, we consider one-dimensional scalar transport equations with oscillatory coefficients. Due to highly oscillatory velocity field, to solve the equation numerically, we must use very fine grid size (order of ϵ) to get reasonable solutions. However, the computational cost can be very heavy. We propose a multiscale method and derive the effected coefficients from microscale equation. We give the error estimate for linear cases and present some numerical experiments. Examples include one- and two-scaled linear problems and the nonlinear flux of Burgers’ equation and Buckley–Leverett equation. Our analysis and numerical experiments show the convergence of our method. Furthermore, the position of shock wave is captured correctly for nonlinear problems.