The Glimm Difference Scheme

Abstract

We consider a general system of conservation laws

$${u_t}\, + \,f{(u)_x}\, = \,0,\,x \in \,R,\,t\, >\,{\text{0,}}$$

(19.1)

where u = (u1,…, u _n ), with initial data

$$u(x,\,0)\, = \,{u_0}(x),\,x \in \,R.$$

(19.2)

The system (19.1) is assumed to be hyperbolic and genuinely nonlinear in each characteristic field, in some open set U ⊂ Rⁿ (see Definition 17.7). We let λ₁(u) < … < λ _n (u) denote the eigenvalues of df(u). Concerning u₀(x), we assume that

$${\text{T}}{\text{.}}\,{\text{V}}{\text{.(}}{u_0}{\text{)}}$$

is sufficiently small, where by T.V.(·) we mean the total variation. With these assumptions, we shall show that the above problem has a solution which exists for all t > 0.