A novel boundary constrained reconstruction method for unstructured finite volume method of Euler equations

For the second-order cell-centered unstructured finite volume method, the variables are linearly distributed inside the grid cell. Classical methods use a first-order extrapolation process in calculating the boundary values, which can lead to loss of second-order accuracy at the boundary. In order to solve the problem, a novel boundary constrained reconstruction method is proposed for boundary values in the unstructured finite volume method. This method first solves for all cell-centered gradients using the weighted least squares method. Subsequently, the boundary cell center gradient is used to reconstruct the boundary face center values. These reconstructed boundary values, corrected by boundary conditions, are then added to the boundary cell gradient calculation stencils to recalculate the boundary cell center gradient. The above-mentioned steps are repeated iteratively, stopping when the boundary face center values stabilize. Numerical case validation shows that this method is able to ensure linear distribution of variables within the boundary cells of the second-order cell-centered unstructured finite volume method, thereby recovering the accuracy of boundary cell calculations.