On Flux-Difference Residual Distribution Methods

Abstract

In this paper, the properties of the newly developed flux-difference residual distribution methods will be analyzed. The focus would be on the order-of-accuracy and stability variations with respect to changes in grid skewness. Overall, the accuracy loss and the stability range of the new methods are comparable with the existing residual distribution methods. It will also be shown that new method has a general mathematical formulation which can easily recover the existing residual distribution methods.

Appendices

A Truncation Error (TE) of Classic RD Methods

The TE for different classic RD methods are determined with the assumption that $\frac{b}{a}<\frac{k}{h}$.

$$\begin{aligned} \text {TE}_\text {N}= & {} \left( -\frac{ab}{2r^2}(ak-bh)\right) u_{nn}\nonumber \\&+\,\left( \frac{ab}{6r^3}(ak-bh)(ak-2bh)\right) u_{nnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=3. \end{aligned}$$

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$$\begin{aligned} \text {TE}_\text {LxF}= & {} \left( -\frac{\left( 4 a^3 k^3-3 a^2 b k^2h+4 a b^2 kh^2+b^3h^3\right) }{6kh\left( a^2+b^2\right) }\right) u_{nn}\nonumber \\&+\,\left( \frac{a b^2 (ak-bh)h}{12 \left( a^2+b^2\right) ^{3/2}}\right) u_{nnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=3. \end{aligned}$$

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$$\begin{aligned} \text {TE}_\text {LDA}= & {} \left( \frac{ab(ak-bh)(ak-2bh)}{6r^3}\right) u_{nnn}\nonumber \\&+\,\left( -\frac{ab^2(ak-bh)^2h}{8r^4}\right) u_{nnnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=4. \end{aligned}$$

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$$\begin{aligned} \text {TE}_\text {LxW}= & {} \left( -\frac{a b^2 (ak-bh)^2h}{12r^4}\right) u_{nnnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=4. \end{aligned}$$

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B Stability Analysis on Classic RD Methods

With the same formulation in Sect. 3.6, the amplification factor, $\delta$ for N and LDA schemes are determined as the following.

$$\begin{aligned} |\delta |^{\mathrm{N}}= & {} \left\{ \left[ 1 + \frac{\nu }{3 s} \left( -s +(s-1) \hbox {cos}(\theta x) + \hbox {cos}(\theta x + \theta y)\right) \right] ^2\right. \nonumber \\&+ \left. \left[ i \frac{\nu }{3 s} \left( (s -1)\hbox {sin}(\theta x) + \hbox {sin}(\theta x + \theta y)\right) \right] ^2\right\} ^{\frac{1}{2}}. \end{aligned}$$

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$$\begin{aligned} |\delta |^{\mathrm{LDA}}= & {} \left\{ \left[ 1 + \frac{\nu }{3 s^2} \left( -s^2+s-1+(s-1)s \hbox {cos}(\theta x)\right. \right. \right. \nonumber \\&\left. \left. +(1-s)\hbox {cos}(\theta y)+ s \hbox {cos}(\theta x + \theta y)\right) \right] ^2\nonumber \\&+ \left. \left[ i \frac{\nu }{3 s} \left( (s -1)\hbox {sin}(\theta x) + \hbox {sin}(\theta x + \theta y)\right) \right] ^2\right\} ^{\frac{1}{2}}. \end{aligned}$$

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