Characteristics-Based, High-Order Accurate and Nonoscillatory Numerical Method for Hyperbolic Heat Conduction

For extremely short durations oral very low temperatures (near absolute zero), the classical Fourier heat conduction equation fails and has to be replaced by a hyperbolic equation to account for finite thermal wave propagation. During the last few years, there has been a growing interest in numerical simulation of the hyperbolic heal conduction problem. The schemes used in previous studies were either the classical upwind and central difference or the MacCormack's predictor-corrector schemes. As a result, spurious oscillations or excessive diffusion appeared near the discontinuities. This paper presents a method that makes use of the characteristics to suppress these oscillations or diffusions. The equations governing the hyperbolic heat conduction problem are first transformed into characteristic equations, and the first-order upwind, second-order upwind (Beam-Warming)and second-order central (Lax-Wendroff)schemes are applied based on the direction of the characteristic velocity. It is shown that simple implementation of these schemes cannot lead to satisfactory results. Subsequently, the high-order TVD (Total Variation Diminishing) schemes are introduced to solve the problem. These TVD schemes are oscillation free and can give high-order accuracy without introducing wiggles. The principle of the proposed scheme is demonstrated by proper switching between Lax-Wendroff and Beam-Warming schemes. Basically, the higher-order scheme is used in the calculation domain, except at discontinuities and local extrema where it is switched to the first-order scheme. Several examples are used to demonstrate the success of the numerical method.