Proc. of 7th Conference of China Numerical Methods on Fluid Mechanics Changchun, China, 1995, p.50

On the Down Flow Boundary, an Extrapolation that Preserves TVD for Out Flow Problems^{1}

YU Xin^{2}, WANG Famin, ZHAO Lie

(Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China)

[First paragraph] —- omitted.

When we solve an out flow problem numerically, we cut the infinite solution domain to a finite scale. One method to decide the value on outflow boundary is extrapolation. Zero gradient extrapolation has a good stability with low order accurate; Higher order accurate extrapolations sometimes lose stability. Now we present an extrapolation (on the boundary) that Preserves TVD for out flow problems. It obtains both higher order accurate and a good stability. It can be extended to other schemes.

For a three point TVD scheme, Set U_{N}^{n+1} on the boundary as

U_{N}^{n+1} = U_{N−1}^{n+1} + sign(Δ_{N− \frac{1}{2}}^{n+1} U)min(|Δ_{N− \frac{1}{2}}^{n+1} U|, |U_{N}^{n} − U_{N−1}^{n+1}|)  (1)

where Δ_{N− \frac{1}{2}}^{n+1} U is a first order accurately (linearly) extrapolated value minus

U_{N−1}^{n+1}, or a second order accurately extrapolated value minus U_{N−1}^{n+1}, or “min mod”^{3} of them, then TVD property is preserved.

Theorem 1. Consider the one dimensional hyperbolic conservation law

\frac{∂U}{∂t} + \frac{∂f(U)}{∂x} = 0  (E)

and a 2k + 1 point finite difference scheme of it:

U_{i}^{n+1} = U_{i}^{n} − Δtf̄(U_{k−1}^{n}, ..., U_{k+1}^{n})  (S)

1Original title is: “An Extrapolation that Preserves TVD for Out Flow Problems”

2E-mail: yu@cerse.psu.edu

Home pages: http://www.cerse.psu.edu/yu/, http://www.yuxin.com/

3min mod(a, b)=sign(a)min(abs(a),abs(b)) when ab > 0, mod(a, b)=0 when ab ≤ 0

�\underline{2} �

Suppose it is TVD, i. e., for any {U_{i}^{n}},

∑_{i=−∞}^{∞} |U_{i+1}^{n+1} − U_{i}^{n+1}| ≤ ∑_{i=−∞}^{∞} |U_{i+1}^{n} − U_{i}^{n}|  (2)

and suppose for any i, U_{i−1}^{n} = U_{i}^{n} = U_{i+1}^{n} implies U_{i}^{n+1} = U_{i}^{n}. Then as along as U_{N}^{n+1}, ..., U_{N+k−1}^{n+1} satisfy

|U_{N}^{n+1} − U_{N−1}^{n+1}| + ... + |U_{N+k−1}^{n+1} − U_{N+k−2}^{n+1}| ≤ |U_{N+k−1}^{n} − U_{N−1}^{n+1}|  (3)

we have the TVD property preserved, i. e.,

∑_{i=−∞}^{N+k−2} |U_{i+1}^{n+1} − U_{i}^{n+1}| ≤ ∑_{i=−∞}^{N+k−2} |U_{i+1}^{n} − U_{i}^{n}|  (4)

Inner point values U_{i}^{n+1}(i ≤ N −1) are calculated by the finite difference scheme. Formulae (for k points U_{N}^{n+1}, ..., U_{N+k−1}^{n+1} on/out of the boundary) satisfying (3) can be given refering to (1).