---
tag: Y70AC2X30T
kind: concept
name: Glimm scheme
msc: [35L65, 65M06, 35L45]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_problem
---

## Summary

The Glimm scheme, or random choice method, builds approximate solutions of a hyperbolic system of [[conservation laws|8RA7AMWSAF]] in one space dimension.
It pieces together exact solutions of [[Riemann problems|GJWKX22VVE]] on a grid and samples them at random points.
Glimm used it in 1965 to prove global existence of weak solutions for initial data of small total variation.
See the [[definition|5JQG4S5JVH]].

## Explanations

### L1 human

Cut the initial data into small steps, so it is constant on each cell.
At each cell boundary the data has a jump, and the solution of a jump is known: it is a Riemann problem.
Run these local solutions for a short time, before they can meet.
Then flatten the result again into steps, taking on each cell the value at one randomly chosen point.
Repeat this many times, and let the grid become finer.