Convergence Study

Q: Why use Convergence Study on TypingMind?

Because you install it once and use it with any model. Convergence Study is plain Markdown rather than provider-specific code, so the same skill runs on GPT-5, Claude, Gemini, Grok, or a local model — and you can switch model mid-chat without it breaking. TypingMind runs on your own API keys, so you pay providers directly instead of a per-seat subscription, and your skills and chats stay in your own storage.

Q: How do I install Convergence Study in TypingMind?

Open Plugins → Skills → Install from GitHub in TypingMind and paste https://github.com/beita6969/ScienceClaw/tree/main/skills/convergence-study. TypingMind reads its SKILL.md and bundles its files and installs it as a skill you can enable per chat.

Q: Which AI models can use Convergence Study?

Any model you connect in TypingMind. AI skills are plain Markdown instructions rather than provider-specific code, so GPT, Claude, Gemini, Grok, and local models can all load this skill when a request matches it.

Q: How many AI models can I use with Convergence Study?

As many as you like. As long as a model supports skills, you can use Convergence Study with it — GPT, Claude, Gemini, Grok, DeepSeek, Mistral, Llama and more — all on TypingMind with your own API keys.

Q: Is the Convergence Study AI skill free?

Yes. It is published on GitHub by beita6969 under the MIT license. You only pay your own AI provider for the tokens you use.

Q: What are AI skills?

An AI skill is a reusable instruction bundle that teaches an AI model how to do one specific task. It follows the open Agent Skills format: a SKILL.md file with a name and description, plus any scripts, templates or reference files the model may need. The model reads the instructions only when your request matches the skill, so an installed skill costs nothing until it is used.

Q: How are AI skills different from plugins or MCP servers?

A plugin or MCP server gives a model new tools to call — code that runs somewhere and returns a result. An AI skill gives the model knowledge and process instead: how to approach a task, which steps to follow, what good output looks like. Skills are plain Markdown, so they need no server, no API key and no runtime, and they work with any model.

Community

beita6969

convergence-study

Spatial and temporal convergence analysis with Richardson extrapolation and Grid Convergence Index (GCI) for solution verification

Overview

Publisher	beita6969
Repository	`ScienceClaw`
Skill name	`convergence-study`
Stars	950
Forks	106
Bundled files	6
License	MIT
Links	Skill source Repository Homepage

Markdown instructions
A SKILL.md file the model loads on demand, so it only costs tokens when a request actually matches.
Works with any LLM
AI skills are plain Markdown, not provider-specific code, so this works with GPT, Claude, Gemini, Grok, or a local model.
6 bundled files
Scripts, templates, and references the model can read while it works. Files are read-only and never executed.
Open source
Published by beita6969 on GitHub. Read the source before you install it.

Installation

Install the Convergence Study AI skill in TypingMind to use it with any LLM, or drop it into another agent that reads SKILL.md.

Install in TypingMind

TypingMind installs a skill straight from its GitHub folder — it reads SKILL.md, bundles the resource files, and stores the result locally.

Open the app and go to Plugins → Skills.
Choose "Install from GitHub".
Paste the skill folder URL below and confirm.
https://github.com/beita6969/ScienceClaw/tree/main/skills/convergence-study
Enable the skill in any chat where you want it available.

Plugins → Skills → Add skill → From GitHub URL, then paste the folder URL and press Continue.

Install in another agent

Any agent that reads the Agent Skills format can use this skill — copy the folder into that agent's skills directory.

Claude Code — .claude/skills

git clone --depth 1 https://github.com/beita6969/ScienceClaw.git /tmp/ScienceClaw
mkdir -p .claude/skills
cp -r /tmp/ScienceClaw/skills/convergence-study .claude/skills/convergence-study

Restart Claude Code after copying so it picks up the new skill.

Use it in TypingMind

Enable Convergence Study in any TypingMind chat and the model takes it from there. Its name and description sit in the system prompt, and the moment a request matches, the model loads the full instructions itself — you never invoke it by hand, and it costs no tokens until it is actually used.

The model loads Convergence Study on its own as soon as a request matches it.

Works with any AI model

AI skills are plain Markdown instructions rather than provider-specific code, so Convergence Study is not tied to the model it was written for. Install it once in TypingMind and use it with GPT-5, Claude, Gemini, Grok, DeepSeek, Mistral, Llama, or a local model you run yourself — all on your own API keys.

Loaded only when it is needed
The system prompt carries just the name and description. The instructions are fetched on the first matching request, so an idle skill costs nothing.
Switch models mid-chat
Because the skill is instructions rather than code, changing model does not break it — the next model reads the same SKILL.md.

Skill instructions

This is the SKILL.md content the model loads. Read it before installing — a skill is instructions your model will follow.

Convergence Study

Goal

Provide script-driven convergence analysis for verifying that numerical solutions converge at the expected rate as the mesh or timestep is refined.

Requirements

Python 3.8+
NumPy (not required; scripts use only math stdlib)

Inputs to Gather

Input	Description	Example
Grid spacings	Sequence of mesh sizes (coarse to fine)	`0.4,0.2,0.1,0.05`
Timestep sizes	Sequence of dt values	`0.04,0.02,0.01`
Solution values	QoI at each refinement level	`1.16,1.04,1.01,1.0025`
Expected order	Formal order of the numerical scheme	`2.0`
Safety factor	GCI safety factor (1.25 default)	`1.25`

Script Outputs (JSON Fields)

Script	Key Outputs
`scripts/h_refinement.py`	`results.observed_orders`, `results.mean_order`, `results.richardson_extrapolated_value`, `results.convergence_assessment`
`scripts/dt_refinement.py`	Same as h_refinement but for temporal convergence
`scripts/richardson_extrapolation.py`	`results.extrapolated_value`, `results.error_estimate`, `results.observed_order`
`scripts/gci_calculator.py`	`results.observed_order`, `results.gci_fine`, `results.gci_coarse`, `results.asymptotic_ratio`, `results.in_asymptotic_range`

Workflow

Run grid/timestep refinement study with at least 3 levels
Compute observed convergence order with h_refinement.py or dt_refinement.py
Compare observed order to expected order of the scheme
Estimate discretization error via Richardson extrapolation
Report GCI for formal solution verification using gci_calculator.py
Document convergence results and any anomalies

Decision Guidance


Do you have 3+ refinement levels?
+-- YES --> Run h_refinement.py or dt_refinement.py
|           +-- Observed order matches expected? --> Solution verified
|           +-- Order too low? --> Check: pre-asymptotic, coding error, insufficient resolution
|           +-- Order too high? --> Check: superconvergence or cancellation effects
+-- NO (only 2 levels) --> Use richardson_extrapolation.py with assumed order
                           (less reliable without order verification)

CLI Examples

bash
# Spatial convergence with 4 grid levels
python3 scripts/h_refinement.py --spacings 0.4,0.2,0.1,0.05 --values 1.16,1.04,1.01,1.0025 --expected-order 2.0 --json

# Temporal convergence with 3 timestep levels
python3 scripts/dt_refinement.py --timesteps 0.04,0.02,0.01 --values 2.12,2.03,2.0075 --expected-order 2.0 --json

# Richardson extrapolation with assumed 2nd-order
python3 scripts/richardson_extrapolation.py --spacings 0.02,0.01 --values 1.0032,1.0008 --order 2.0 --json

# GCI for 3-mesh verification
python3 scripts/gci_calculator.py --spacings 0.04,0.02,0.01 --values 1.0128,1.0032,1.0008 --json

Error Handling

Error	Cause	Resolution
`spacings and values must have the same length`	Mismatched input arrays	Provide equal-length lists
`At least 2 refinement levels required`	Too few data points	Add more refinement levels
`Exactly 3 refinement levels required`	GCI needs 3 levels	Provide fine/medium/coarse
`Oscillatory convergence detected`	Non-monotone convergence	Check mesh quality or scheme

Interpretation Guidance

Scenario	Meaning	Action
Observed order matches expected	Solution in asymptotic range	Report GCI, extrapolate
Observed order < expected	Pre-asymptotic or coding bug	Refine further or debug
Negative observed order	Solution diverging	Check implementation
GCI asymptotic ratio near 1.0	Grids in asymptotic range	Results are reliable
GCI asymptotic ratio far from 1.0	Not in asymptotic range	Refine further

References

references/convergence_theory.md - Formal convergence order, log-log analysis, asymptotic range
references/gci_guidelines.md - Roache's GCI method, ASME V&V 20, safety factors

Bundled files

The model reads these on demand while the skill is loaded. They are exposed as readable files and are never executed.

Frequently asked questions

What does the Convergence Study AI skill do?

Spatial and temporal convergence analysis with Richardson extrapolation and Grid Convergence Index (GCI) for solution verification

Why use Convergence Study on TypingMind?

Because you install it once and use it with any model. Convergence Study is plain Markdown rather than provider-specific code, so the same skill runs on GPT-5, Claude, Gemini, Grok, or a local model — and you can switch model mid-chat without it breaking. TypingMind runs on your own API keys, so you pay providers directly instead of a per-seat subscription, and your skills and chats stay in your own storage.

How do I install Convergence Study in TypingMind?

Open Plugins → Skills → Install from GitHub in TypingMind and paste https://github.com/beita6969/ScienceClaw/tree/main/skills/convergence-study. TypingMind reads its SKILL.md and bundles its files and installs it as a skill you can enable per chat.

Which AI models can use Convergence Study?

Any model you connect in TypingMind. AI skills are plain Markdown instructions rather than provider-specific code, so GPT, Claude, Gemini, Grok, and local models can all load this skill when a request matches it.

How many AI models can I use with Convergence Study?

As many as you like. As long as a model supports skills, you can use Convergence Study with it — GPT, Claude, Gemini, Grok, DeepSeek, Mistral, Llama and more — all on TypingMind with your own API keys.

Is the Convergence Study AI skill free?

Yes. It is published on GitHub by beita6969 under the MIT license. You only pay your own AI provider for the tokens you use.

What are AI skills?

An AI skill is a reusable instruction bundle that teaches an AI model how to do one specific task. It follows the open Agent Skills format: a SKILL.md file with a name and description, plus any scripts, templates or reference files the model may need. The model reads the instructions only when your request matches the skill, so an installed skill costs nothing until it is used.

How are AI skills different from plugins or MCP servers?

A plugin or MCP server gives a model new tools to call — code that runs somewhere and returns a result. An AI skill gives the model knowledge and process instead: how to approach a task, which steps to follow, what good output looks like. Skills are plain Markdown, so they need no server, no API key and no runtime, and they work with any model.

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Overview

Installation

Install in TypingMind

Install in another agent

Use it in TypingMind

Works with any AI model

Skill instructions

Convergence Study

Goal

Requirements

Inputs to Gather

Script Outputs (JSON Fields)

Workflow

Decision Guidance

CLI Examples

Error Handling

Interpretation Guidance

References

Bundled files

Frequently asked questions

Related AI Skills

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