מושגי ליבה
The spectral difference method based on the Raviart-Thomas space (SD-RT) exhibits different convergence rates on regular triangular meshes depending on the alignment of the transport velocity with the mesh edges.
תקציר
The paper analyzes the accuracy of the SD-RT method for solving the transport equation on regular triangular meshes. The key findings are:
- If the transport velocity is parallel to a family of mesh edges, the SD-RT(p) method converges with order p.
- If the transport velocity is not parallel to any mesh edges, the SD-RT(p) method converges with order p+1.
This behavior is proved theoretically for p=1 and demonstrated numerically for p=1, 2, 3. The analysis relies on studying the properties of the scheme's truncation error and its relation to the co-kernel of the scheme's matrix.
The paper also shows that for the case where the velocity is not parallel to the mesh, the SD-RT(1) scheme achieves second-order accuracy in the long-time simulation.
סטטיסטיקה
The transport velocity ω = (ωx, ωy)^T is constant.
The initial data v0 is sufficiently smooth and periodic with the periodic cell (0, 1)^2.
ציטוטים
"If ωx, ωy > 0, then the optimal order of accuracy is 2. If ωx > ωy = 0, then the optimal order of accuracy is 1."
"For t ≫ 1/h the distance between lines corresponding to different h is identical for both cases and corresponds to the (p+1)-th order convergence."