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The explicit Euler, midpoint, and classical fourth-order Runge–Kutta tableaus
Definition
The following are the explicit Runge–Kutta tableaus in the convention of Explicit Runge–Kutta tableaus, stage maps, and failure status.
Forward Euler has , , , and .
Explicit midpoint has , , , and .
The classical four-stage Runge–Kutta method has , , , and .
Each displayed is the row sum required in the definition of an explicit tableau. These tableaus specify the stage equations; their order and scalar stability functions are proved separately.
Depends on
Used by
- Classical RK4 step for the fixed-grid regularized PIC particle ODE Algorithm
- A dissipative scalar heat mode with Parareal correction Example
- A scalar diffusion mode gives an explicit two-grid MGRIT matrix Example
- Explicit midpoint loses its uniform second-order defect bound at a time kink Example
- Fixed-time error slopes for Euler, midpoint, and classical RK4 Example
- One-step Lipschitz bound for classical RK4 on an admitted tube Lemma
- Orders and scalar stability functions of Euler, midpoint, and classical RK4 Proposition
- Fourth-order global error for RK4 on a regularized vortex particle tube Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)
- MIT OpenCourseWare, 18.330 Introduction to Numerical Analysis, Chapter 5: Methods for Ordinary Differential Equations (standard reference, not scraped)