Operator Splitting for Convection-Dominated Nonlinear Partial Differential Equations

Abstract

We describe an efficient solution strategy for nonlinear systems of partial differential equations of the form

$$ {U_t} + \sum\limits_i {{F_i}} {(U)_{{x_i}}} = \sum\limits_{i,j} {{D_{ij}}} {(U)_{{x_i}{x_j}}} + G(U),{\text{ U}}{{\text{|}}_{t = 0}} = {U_0}. $$

(1)

We explicitly allow for degeneracy of the viscous term in the sense that we only require Σ_i,jD′_ij (u) ξ_iξ_j ≥ 0. The solution strategy is based on operatoi splitting where an abstractly defined Cauchy problem U_t + A(U) = 0, is split into simpler problems V ^l_t + A^l(V^l) = 0, by writing A = A¹ +…+ A^ℓ. If the solution of subproblem I is written V^l(t) = S ^l_t V ^l₀ then the idea is thai an approximate solution of the original problem reads

$$ U\left( {n\Delta t} \right) \approx {U^n} = {\left[ {S_{\Delta t}^\ell\circ\cdots\circ S_{\Delta t}^1} \right]^n}{U^0},{\text{ }}t = n\Delta t. $$