文档介绍：(六B)双曲型守恒律及可压缩流的高分辨率格式
TVD, NND, MUSCL,ENO,WENO, 群速度控制法,Godunov, Roe, CESE, DG…
TVD格式(续)
MUSCL格式
NND格式
群速度控制法
WENO格式
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TVD格式(续)
带通量限制子的二阶TVD
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单调格式(monotone scheme)
A scheme
with non-negative integers, is said to be monotone if
That is, H is a non-decreasing function of each of its arguments
Example: 1st order upwind scheme
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TVD 格式的定义
is said to be Total Variation Diminishing (TVD) if
单调保持格式的定义 Monotonicity Preserving (MP) Scheme. A scheme is said to be MP if whenever the data is monotone the solution is monotone in the same sense.
Theorem (Harten)
Lecture 6
That is: monotone schemes and TVD and TVD schemes are monotonicity preserving schemes
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带通量限制子的TVD
To guarantee 2nd order accuracy and avoid pression of solutions, Sweby suggested the following TVD region as a suitable range for the flux limiting function:
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Minmod Flux Limiter on Sweby Diagram
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It is apparent that the minmod flux limiter applies the maximum possible limiting allowed within the second order TVD region.
(. it will be rather dissipative and smear out discontinuities somewhat as seen on the right hand side figure).
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Superbee Flux Limiter on Sweby Diagram
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The Superbee limiter applies the minimum limiting and maximum steepening possible to remain TVD. It is known to suffer from excessive sharpening of slopes as a result.
On the right we show what happens to a smooth sine wave after 20 periods.
Notice the flattening of the peaks and the steepening of the slopes.
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MC Flux Limiter on Sweby Diagram
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The MC limiter transitions from upwind (theta<0) to Fromm (at theta=1/3) then switches to a constant(at theta=3).
This is promise between Superbee and minmod
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van Leer Flux Limiter
The van Leer limiter charts a promise path through the Sweby TVD region.
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Summary of Some Flux Limiting Functions
Nonlinear second order TVD limiters
Linear non-TVD limiters
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