時空守恆元解元方法應用於保守律的數值研究

Abstract

英文摘要

In this thesis, the main task is to solve conservation laws with a discontinuous flux function in space applying the space time conservation elements and solution elements method (CESE). CESE is an explicit method with accuracy of second order at least. We will review the key idea of CESE by considering an simple linear case. Next, CESE method for conservation laws with non-linear flux will be derived and applied to discontinuous flux function. On the interface, the basic strategy is to apply the Newton's method. Also, the Steffensen's method is applied for avoiding taking complicated derivatives. Our new modified CESE method is accurate of order 2 at least in $L_1$ and $L_2$ norms. Then, we perform the modified CESE method for the cases of discontinuous flux function, $f$ and $g$, which satisfy the assumption that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. Several numerical examples with Riemann problems are also performed and presented.