1 Introduction
2 Upwind Discretization of the Advection Equation
2.1 New First Order Upwind SBP Operators
2.2 Convergence to Steady-State for Single Grid Methods
3 Convergence Acceleration for First Order Upwind Schemes
- A coarse grid \(\varOmega _{2} = \left\{ x_{j}^{(2)} = 2j \varDelta x^{(1)} = j \varDelta x^{(2)}, \ j = 0, 2, \ldots , N\right\} \)\(\subset \)\(\left[ 0,1\right] \);
- A restriction operator of injection type \(\mathcal {R}_{u}\) such that \(\left( \mathcal {R}_{u} \varvec{v}^{(1)}\right) _{j} = v^{(2)}_{j}\) for \(j = 0, 2, \ldots , N\);
- A residual restriction operator \(I_{r}\) which conveys the information from \(\varOmega _{1}\) to \(\varOmega _{2}\),which is upwind-biased [4, 5] at interior nodes and inconsistent at the left boundary node. This somewhat odd choice will be explained later. See Fig. 3 for details;$$\begin{aligned} \left( I_{r} \varvec{v}^{(1)} \right) _{j} = {\left\{ \begin{array}{ll} \begin{array}{lll} \frac{1}{2} v^{(1)}_{0}, &{}\quad &{}j = 0, \\ \frac{1}{2} \left( v_{j}^{(1)} + v_{j-1}^{(1)}\right) , &{} &{}j = 2, 4, \ldots , N, \end{array} \end{array}\right. } \end{aligned}$$(12)
- The prolongation operator \(I_{p} = I_{p}^{I} + I_{p}^{E}\) in the fine grid update step [12] has been split into two contributions: \(I_{p}^{I}\) for the nodes included on the coarse grid and \(I_{p}^{E}\) for the other nodes, see Fig. 3. In particular,are introduced to avoid overshoots in the transient phase, and preserve the TVD property [4].$$\begin{aligned} \begin{array}{l} \left( I_{p}^{I} \varvec{v}^{(2)} \right) _{j} = {\left\{ \begin{array}{ll} \begin{array}{l@{\quad }l} v_{j}^{(2)}, &{} j = 0, 2, 4, \ldots , N, \\ 0, &{} j = 1, 3, 5, \ldots , N-1, \end{array} \end{array}\right. } \\ \left( I_{p}^{E} \varvec{v}^{(2)} \right) _{j} = {\left\{ \begin{array}{ll} \begin{array}{ll} 0, &{} j = 0, 2, 4, \ldots , N, \\ v_{j-1}^{(2)}, &{} j = 1, 3, 5, \ldots , N-1, \end{array} \end{array}\right. } \end{array} \end{aligned}$$(13)
3.1 Initial Modifications for Higher Order Discretizations
3.2 Revisiting the Upwind-Biased Residual Restriction
4 Extension to Higher Orders of Accuracy
4.1 Prolongation Operators
4.2 Second Order Discretizations
4.3 Extension to Higher Orders
\(N{\backslash }\)Discretization | 1st | 2nd | 3rd | 4th | 5th | 6th |
---|---|---|---|---|---|---|
100 | 7.520e−2 | 2.130e−2 | 2.474e−3 | 3.972e−4 | 2.652e−4 | 2.664e−4 |
200 | 3.785e−2 | 5.563e−3 | 4.134e−4 | 2.303e−5 | 1.412e−5 | 1.430e−5 |
300 | 2.531e−2 | 2.500e−3 | 1.469e−4 | 4.449e−6 | 2.683e−6 | 2.395e−6 |
400 | 1.901e−2 | 1.414e−3 | 7.081e−5 | 1.404e−6 | 8.526e−7 | 6.654e−7 |
500 | 1.522e−2 | 9.072e−4 | 4.027e−5 | 5.789e−7 | 3.569e−7 | 2.455e−7 |
Order | 0.9965 | 1.9889 | 2.5293 | 3.9703 | 3.9026 | 4.4684 |
Expected | 1 | 2 | 2 | 3 | 3 | 4 |
4.4 The SBP-Preserving Prolongation
5 Extension to Hyperbolic Systems of Equations
5.1 Modifications of the Multigrid Procedure for Systems of Equations
5.2 Numerical Results for Characteristic Boundary Conditions
5.3 Numerical Results for Other Boundary Conditions
6 Extension to Nonlinear Problems
6.1 Interpolation Operators for the Nonlinear Case
\(N{\backslash }\) Discretization | 1st | 2nd | 3rd | 4th | 5th | 6th |
---|---|---|---|---|---|---|
100 | 2.394e−2 | 6.892e−5 | 3.992e−5 | 8.254e−7 | 7.982e−7 | 2.065e−8 |
200 | 1.165e−2 | 1.741e−5 | 9.920e−6 | 1.038e−7 | 1.004e−7 | 1.308e−9 |
300 | 7.702e−3 | 7.763e−6 | 4.400e−6 | 3.082e−8 | 2.982e−8 | 2.596e−10 |
400 | 5.752e−3 | 4.374e−6 | 2.473e−6 | 1.302e−8 | 1.259e−8 | 8.238e−11 |
500 | 4.590e−3 | 2.802e−6 | 1.581e−6 | 6.668e−9 | 6.452e−9 | 3.381e−11 |
Order | 1.0113 | 1.9954 | 2.003 | 2.997 | 2.997 | 3.991 |
Expected | 1 | 2 | 2 | 3 | 3 | 4 |