1.1 Overview of Hyperbolic Partial Differential Equations

The important concepts of convergence, consistency, and stability are presented and shown to be related by the Lax-Richtmyer equivalence theorem. The chapter concludes with a discussion of the Courant-Friedrichs-Lewy condition and related topics.

Overview of Hyperbolic Partial Differential Equations

The One-Way WaveEquation

$$ u_t + au_x = 0 \label{1.1.1}\tag{1.1.1}$$ solution: $$ u(t,x) = u_0 ( x - at) \tag{1.1.2}$$ \((t,x)\) plane에서 \(x-at\)가 상수로 유지되는 라인을 characteristics라고 부른다. \(a\)는 the speed of propagation along the characteristic.

One-way wave eq.(식 (1))의 solution(식 (2))은 형태의 변형 없이 speed \(a\)로 진행하는 wave이다.

식(2)는 미분가능성을 요하지 않는다.

$$ u_t + au_x + bu = f(t, x),\\u(0, x) = u_0(x), \tag{1.1.3} $$ $$ u(t,x) = u_0(x-at)e^{-bt} + \int_0^t f(s, x-a(t-s))e^{-b(t-s)} ds. \tag{1.1.4}$$ $$ u_t + au_x = f(t,x,u) \tag{1.1.5}$$

Systems of Hyperbolic Equations

$$ u_t + Au_x + Bu = F(t, x) \tag{1.1.6}$$