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

Data Analysis

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

Data Analysis

Comparison

Unstructured Grid

20 x 20 numerical cells

Unstructured Grid

40 x 40 numerical cells

Unstructured Grid

80 x 80 numerical cells

Data Analysis

Comparison

Polygon Grid

20 x 20 numerical cells

Polygon Grid

40 x 40 numerical cells

Polygon Grid

80 x 80 numerical cells

Data Analysis

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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