Coupled sound and heat flow and the method of least squares

Abstract:

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain $\Omega$ in ${R^N}$, with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-squares method is devised, using trial functions which need not satisfy the homogeneous boundary conditions. Certain unknown normal derivatives of the solution enter the boundary integrals. By using suitable weights, these unknown derivatives can be set equal to zero without impairing the $O({k^2})$ accuracy of the Crank-Nicolson scheme. However, one must use smoother trial functions to obtain this accuracy.