Difference Methods for One-Dimensional PDE

Abstract

Solving partial differential equations (PDE) is so crucial to mathematical physics that we devote three chapters to it. The solution methods largely depend on the type of the PDE. The type of a general system of linear first-order equations

$$ A{{\varvec{v}}}_x + B{{\varvec{v}}}_y = {{\varvec{c}}} \>, $$

where ${{\varvec{v}}}=(v_1, v_2, \ldots , v_M)^\mathrm {T}$ is the solution vector, ${{\varvec{c}}}=(c_1, c_2, \ldots , c_M)^\mathrm {T}$ is the vector of inhomogeneous terms, and A and B are $M\times M$ matrices, is determined by the zeros of the characteristic polynomial $\rho (\lambda )=\det (A-\lambda B)$. The system is hyperbolic if $\rho (\lambda )$ has precisely M distinct zeros, or if it has M real zeros and the system $(A-\lambda B){{\varvec{u}}}={{\varvec{0}}}$ possesses precisely M linearly independent solutions. The system is parabolic if $\rho (\lambda )$ has M real zeros but $(A-\lambda B){{\varvec{u}}}={{\varvec{0}}}$ does not have M linearly independent solutions. The system without real zeros of $\rho (\lambda )$ is elliptic. (The classification of systems with mixed real and complex zeros of $\rho (\lambda )$ is more involved.)