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Intermediate Time Levels in a Three-Step ADI Method

Article Quant Q&A · Author: A.Oreo
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Summary

The document asks what the fractional time indices in an alternating direction implicit (ADI) numerical scheme represent. Its example advances a quantity through three substeps within a full time step and uses different time levels for the spatial operators in each coordinate direction. The asker wonders whether these fractional-index values are averages, like the midpoint level sometimes defined as the average of the current and next full-step values.

The question frames the fractional levels as possible intermediate variables, but the document contains no answer or derivation confirming their precise definitions. In an ADI scheme, such levels commonly denote intermediate states produced sequentially by the substeps, rather than a prescribed average; their exact meaning depends on the specific scheme and its update equations. The excerpt is therefore useful as a numerical-methods question, but it offers no worked solution or evidence and does not connect the method to a particular trading application.

Key ideas

  • The example divides a full time step into three fractional substeps.
  • Different spatial operators use values at different intermediate time levels.
  • The asker distinguishes these levels from a midpoint value defined as an average.
  • The excerpt provides no derivation, and precise intermediate-state definitions depend on the ADI scheme.

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StatisticsDerivatives pricing
Full text
# what is the meaning of $U^{n+1/3}$ ADI method


# what is the meaning of $U^{n+1/3}$ ADI method












For the ADI in numerical method

$$\frac{U^{n+1/3}-U^n}{k/3} = \Delta^2_x U^{n+1/3} + \Delta^2_y U^n + \Delta^2_z U^{n+2/3}$$ $$....$$ $$....$$

don't like $U^{n+1/2} = \dfrac 1 2 (U^{n+1}+U^n),$ I can't find the definition of $U^{n+1/3},U^{n+2/3},$ can some one tell me he definition or it is just a intermedia variables?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.