# The Analysis of OpenFOAM® Scheme

The following article is related to the investigation of numerical discretization schemes available in the OpenFOAM® software. As OpenFOAM® serves more than 30 discretization schemes for the advection term, only a few are mentioned here. The numerical investigation is related to the description gieven in »Equations and Condition«. The presented results are related to the OpenFOAM® version 5. It is worth to mention that the results might be different in newer versions since algorithms, numerics, or C++ coding might be improved.

## General Remark for the LimitedLimitedLinear Scheme

First-to-Second-Order Accuracy

The Limited scheme (LimitedLinear) calculates a limiter that is applied to the field to avoid non-physical values, compare J.H. Ferziger and M. Perić. The numerical analysis below demonstrates that the applied limiter introduces the stability. This can be directly proved if the Linear scheme is used for the calculation. The calculated values of Φ are in the range of the defined interval [0:1]. Due to numerical errors, it is possible to get quantitative lower or higher values than the threshold. The order of magnitude of the error is in the range of 10-e13 (in this example). The comparison of this study shows that one should prefer the usage of limited schemes rather than the pure Linear scheme especially if the mesh is non-orthogonal. The LimitedLinear scheme blends between the Upwind and Linear scheme based on the applied limiter. As the limiter tends toward 1, the scheme is equal to the Upwind discretization.

## The Results of the LimitedLimitedLinear Scheme

### Structured Grid (0°)

20 x 20 numerical cells

### Structured Grid (0°)

40 x 40 numerical cells

### Structured Grid (0°)

80 x 80 numerical cells

Comparison

### Structured Grid (45°)

20 x 20 numerical cells

### Structured Grid (45°)

40 x 40 numerical cells

### Structured Grid (45°)

80 x 80 numerical cells

Comparison

### Unstructured Grid

20 x 20 numerical cells

### Unstructured Grid

40 x 40 numerical cells

### Unstructured Grid

80 x 80 numerical cells

Comparison

### Polygon Grid

20 x 20 numerical cells

### Polygon Grid

40 x 40 numerical cells

### Polygon Grid

80 x 80 numerical cells

Comparison

### Grid Comparison

20 x 20 numerical cells

### Grid Comparison

40 x 40 numerical cells

### Grid Comparison

80 x 80 numerical cells