A high-order finite volume method for solving one-dimensional convection and diffusion equations

A spatially high-order finite volume method for solving convection and diffusion equations is developed and tested in this work. The method performs a high-order piecewise polynomial reconstruction of the local flow field based on the relationship between Taylor's series expansion and the volume-averaged flow quantities. A 5x5 matrix inversion for each cell is done to compute the cell-center variables and derivatives up to fourth order. While a fixed symmetric grid stencil is maintained in smooth flow regions, a detector for large change in linear data slopes is developed to trigger the use of ENO stencil around flow discontinuities. Regular time integration scheme such as the four-stage Runge-Kutta method or the Euler implicit method is used for time integration. The present finite volume method is shown to be spatially fifth-order accurate for the linear convection equation, fourth-order accurate for the linear diffusion equation, and fourth-order accurate for the linear convection-diffusion equation. The shocks captured in solving the inviscid Burger's equation are sharp and oscillation free. For the system of Euler equations, a characteristic limiter is further developed to limit the growth of total variation of the solution. Test examples solving shock-tube problems and the interactions of two blast waves show that various flow discontinuities are captured sharply without spurious oscillations.