Mehmet Alp Üreten, M.Sc.
Department of Scientific Computing
September 2018, 97 pages

Supervisor: Hamdullah Yücel (Institute of Applied Mathematics, Middle East Technical University, Ankara)
Co-supervisor: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)

Abstract

Differential equations are the primary tool to mathematically model physical phenomena in industry and natural science and to gain knowledge about its features. Deterministic differential equations does not sufficiently model physically observed phenomena since there exist naturally inevitable uncertainties in nature. Employing random variables or processes as inputs or coefficients of the differential equations yields a stochastic differential equation which can clarify unnoticed features of physical events. Korteweg-de Vries (KdV) equation with the random input data is a funda- mental differential equation for modeling and describing solitary waves occurring in nature. It can be represented by employing time dependent additive randomness into its forcing or space dependent multiplicative randomness into derivative of the solution. Since analytical solution of the differential equation with the random data input does not exist, quantifying and propagating uncertainty employed on the differential equation are done by numerical approximation techniques. This thesis will focus on numerical investigation of the Korteweg-de Vries equation with random input data by employing stochastic Galerkin in probability space, local discontinuous Galerkin method in spatial dimension, and theta (weighted average) method in temporal dimension. In numerical implementations, both additive noise and multiplicative noise cases are considered by comparing with other numerical techniques such as Monte Carlo and stochastic collocation methods for the probability space and finite difference method for the spatial discretization.

Keywords: uncertainty quantification, stochastic Galerkin method, Korteweg-de Vries equation, local discontinuous Galerkin method