# The Linear Scheme

The following article is related to the Linear scheme, implemented in OpenFOAM®. The article will describe the implementation to understand the usage and how the code works in that particular area. The Linear Scheme is related to the  surface Interpolation schemes. Thus, in OpenFOAM®, the Linear scheme is NOT derived by the limited scheme class.

## General Remarks

The Linear Scheme (Second Order Scheme)

The numerical scheme named Linear gives one the ability to reach a second order accurate solutions. However, this scheme is highly unstable and calculates non-physical values till the solver blows up and crashes. That means, the calculated value of the scalar Φ can be outside of the physical range. In the following case the range is fixed and has to be in the range of [0:1]. In the result section, one can see that some points are lower than zero and higher than one. In the case of an unstructured grid the scheme is unstable and diverges. The values of the scalar grow and blow up (demonstrated in the video). It is well known that the Linear scheme should be avoided for example in the convection term. To avoid non-physical values and diverging solutions, other numerical schemes should be used; J.H. Ferziger and M.Perić.

## The Results

Of The Numerical Analysis

### 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

### Transient Analysis

Divergence Situation

## The Code

Linear scheme

In Progress

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