Space-time Generalized Riemann Problem Solvers of Order k for Linear Advection with Unrestricted Time Step

Abstract

This work concerns high-order approximations of the linear advection equation in very long time. A GRP-type scheme of arbitrary high-order in space and time with no restriction on the time step is developed. In the usual GRP solvers, we consider a polynomial approximation of the solution in space in each cell at the initial time. Here, we add a second polynomial approximation of the solution in time in each interface. Thanks to this double approximation, the resulting scheme is compact. It is proved to be of order k+1 in space and time, where k is the degree of the polynomials. Thanks to the compactness of the scheme, a two-dimensional extension is detailed on unstructured meshes made of triangles. Several numerical test-cases and comparison with existing methods illustrate the excellent behaviour of the scheme.

Appendix

Evolution Step

Case two targets (explicit):

• ∘ We know the solution $u_{i}^{n}(x,y)$ in the cell T _i at time t ⁿ and the solution $v_{j3}^{n+\frac{1}{2}}(t,\omega)$ on the interface $I_{j3}^{n+\frac{1}{2}}$ .

• ∘ We define θ ₁ as the intersection between the characteristic line coming from $s_{j3}^{n}$ and the segment $s_{j1}^{n+1}$. We also define θ ₂, the intersection between the characteristic line coming from $s_{j3}^{n}$ and the segment $s_{j2}^{n+1}$. Then we call M ₁ the projection of θ ₁ on $s_{j1}^{n}$ and M ₂ the projection of θ ₂ on $s_{j2}^{n}$ (see Fig. 21).

  Fig. 21

  

  Incoming solutions on the interface (left) and on the cell (right) for the explicit case two targets

• ∘ The exact solution on the cell T _i at time t ⁿ⁺¹ is:

  

  with:

  

• ∘ The exact solution on the interface $I_{j1}^{n+\frac{1}{2}}$ is:

  

  with:

  

• ∘ The exact solution on the interface $I_{j2}^{n+\frac{1}{2}}$ is:

  

  with:

  

Case one target (explicit):

• ∘ We know the solution $u_{i}^{n}(x,y)$ in the cell T _i at time t ⁿ and the solutions $v_{j1}^{n+\frac{1}{2}}(t,\omega)$ and $v_{j2}^{n+\frac{1}{2}}(t,\omega)$ on the interfaces I _j1 and $I_{j2}^{n+\frac{1}{2}}$.

• ∘ We define θ ₁ as the intersection between the characteristic line coming from the segment $s_{j1}^{n}$ and the segment $s_{j3}^{n+1}$ (i.e. such that $d(s_{j3}^{n},s_{j1}^{n}) = a \Delta t$). We also define θ ₂ as the intersection between the characteristic line coming from the segment $s_{j2}^{n}$ and the segment $s_{j3}^{n+1}$ (i.e. such that $d(s_{j3}^{n},s_{j2}^{n}) = a \Delta t$). Then we call M ₁ the projection of θ ₁ on $s_{j3}^{n}$ and M ₂ the projection of θ ₂ on $s_{j3}^{n}$ (see Fig. 22).

  Fig. 22

  

  Incoming solutions on the interface (left) and on the cell (right) for the explicit case 1

• ∘ The exact solution on the cell T _i at time t ⁿ⁺¹ is:

  

  with:

  

• ∘ The exact solution on the interface $I_{j3}^{n+\frac{1}{2}}$ is:

  

  with:

  

Case two targets (implicit):

• ∘ We know the solution $u_{i}^{n}(x,y)$ in the cell T _i at time t ⁿ and the solution $v_{j3}^{n+\frac{1}{2}}(t,\omega)$ on the interface $I_{j3}^{n+\frac{1}{2}}$ .

• ∘ We define θ ₃ as the intersection between the characteristic line coming from $s_{j3}^{n}$ and the segment [t ⁿ,t ⁿ⁺¹] for (x,y)=(x _j3,y _j3) (see Fig. 23).

  Fig. 23

  

  Incoming solutions on the interface (left) and on the cell (right) for the implicit case 2

• ∘ The exact solution on the cell T _i at time t ⁿ⁺¹ is:

  

  with:

  

• ∘ The exact solution on the interface $I_{j1}^{n+\frac{1}{2}}$ is:

  

  with:

  

• ∘ The exact solution on the interface $I_{j2}^{n+\frac{1}{2}}$ is:

  

  with:

  

Projection Step

The exact solutions are respectively projected onto the spaces $\mathbb {P}^{k}_{C}$ and $\mathbb {P}^{k}_{I}$: in the cell T _i at time t ⁿ⁺¹, the exact solution is projected in $\mathbb {P}^{k}_{C}$, and on the interfaces $I_{j}^{n+\frac{1}{2}}$, the solution is projected in $\mathbb {P}^{k}_{I}$. For the case n^∘1 explicit, this leads to the two minimization problems:

(20)

(21)

for the case n^∘2 explicit, it leads to the three minimization problems:

(22)

(23)

(24)

and for the case explicit two targets, it also leads to three minimization problems:

(25)

(26)

(27)

Thanks to the Petrov-Galerkin conditions, we can rewrite (20), (21) as:

(28)

(29)

and (22)–(24) as:

(30)

(31)

(32)

Using the decomposition of the solutions in their corresponding basis:

the minimization problems (28), (29) are equivalent to solving the two following linear systems of size $\frac{(k+1)(k+2)}{2}$:

and (30)–(32) are equivalent to solving three linear systems of size $\frac{(k+1)(k+2)}{2}$:

The matrices are the same that in the case n^∘1 implicit and the case n^∘2 explicit. For the right hand sides, we have first for the case n^∘1 explicit:

then for the case n^∘2 implicit:

and for the case explicit two targets:

where the coefficients are given by: