A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

Abstract

We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection–diffusion equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields unconditional stability with respect to the Péclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection–diffusion and viscous Burgers’ equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.

A Alternative Discretizations on the Semi-Infinite Subdomain

We summarize here the results presented in [29] on the analysis of various Laguerre-based discretizations of the advection–diffusion equation with constant coefficients on ${\mathbb R}^+=[0,+\infty )$. For the purpose of deriving some discretizations, it can be helpful to reformulate Eq. (39), which we report here for convenience,

$$\begin{aligned} \frac{\partial c}{\partial t}+u\frac{\partial c}{\partial z}=\mu \frac{\partial ^2 c}{\partial z^2} \end{aligned}$$

(66)

as a system of first order equations

$$\begin{aligned} \begin{aligned}&\frac{\partial c}{\partial t} - \mu \frac{\partial v}{\partial z} + uv =0 \\&\frac{\partial c}{\partial z} - v = 0. \end{aligned} \end{aligned}$$

(67)

We assume that solutions vanish at infinity

$$\begin{aligned} \lim _{z\rightarrow +\infty }c(z,t)=0 \end{aligned}$$

(68)

and that either Dirichlet boundary conditions

$$\begin{aligned} c(0,t) = c_L \end{aligned}$$

(69)

or Neumann boundary conditions

$$\begin{aligned} \frac{\partial c}{\partial z}(0,t) = Dc_L \end{aligned}$$

(70)

are applied at $z=0.$ We require that $\mu >0$ (ellipticity condition) and $u>0$. In this case, the Dirichlet datum at $z=0$ corresponds to an inflow boundary condition, which guarantees well-posedness for the hyperbolic part. We analyze several possible space discretizations, in order to determine which one shows the best stability properties and can therefore be chosen for the extended DG scheme in conjunction with the Legendre basis in the finite sub-domain. As done in [7] for the pure advection problem, we discretize the PDE system (67) in space, obtaining, after substitution of the discretization of the second equation in (67) into the first, a system of ordinary differential equations of the form

$$\begin{aligned} \frac{d{\mathbf {c}}}{dt}={\mathbf {A}}{\mathbf {c}}+{\mathbf {g}}, \end{aligned}$$

(71)

where ${\mathbf {c}}$ is the unknown vector of the expansion of the solution and ${\mathbf {g}}$ contains the contribution of boundary conditions at $z=0$, and we study the eigenvalue structure of the matrix ${\mathbf {A}}$. The corresponding discretization scheme is stable if all the eigenvalues have non-positive real part.

We analyse the following discretizations:

• Weak form We multiply (67) by a test function, integrate by parts and use either Gauss-Laguerre-Radau (GLR) or Gauss-Laguerre (GL) quadrature rules. Two different approaches are possible. In a modal approach, entries of the unknown vector ${\mathbf {c}}$ are the coefficients of the expansion of the solution in the orthogonal basis of Laguerre functions or Laguerre polynomials. In a nodal approach, the basis functions are the Lagrange basis functions associated with the integration nodes, so that the unknown vector contains the nodal values of the approximate solution. Furthermore, the numerical solution can be expanded in a basis of either scaled Laguerre functions or scaled Laguerre polynomials.

• Strong form In this case we directly discretize the strong formulation (67) using a collocation approach and GLR quadrature rules. This is the only practical choice if Dirichlet boundary conditions have to be imposed, because the GLR nodes include the left endpoint of the semi-infinite subdomain, unlike the GL nodes.

We now summarize some definitions we need to introduce the different variants of the matrix ${\mathbf {A}}$ and vector ${\mathbf {g}}.$ For discretizations based on Laguerre functions, we define the matrix $\hat{{\mathbf {L}}}=\{\hat{l}_{ij}\}$ with entries such that

$$\begin{aligned} \hat{l}_{ij}={\left\{ \begin{array}{ll} 1/2 &{} i=j \\ 1 &{} j<i \\ 0 &{} j>i. \end{array}\right. } \end{aligned}$$

(72)

If discretizations based on Laguerre polynomials are considered, we use the matrix ${\mathbf {L}}=\{l_{ij}\}$ defined  as

$$\begin{aligned} l_{ij}={\left\{ \begin{array}{ll} 0 &{} i=j \\ 1 &{} j<i \\ 0 &{} j>i \end{array}\right. } \end{aligned}$$

(73)

For nodal discretizations based on the weak form and on scaled Laguerre functions, we then denote by $z_j^\beta$ the j-th GLR or GL quadrature node, by $h_j^\beta (z)$ the associated Lagrangian polynomial, by $\omega _i$ the i-th quadrature weight, and by $\hat{d}_{ij}^\beta$ the entries of the GLR or GL differentiation matrix $\hat{{\mathbf {D}}}_\beta$ associated with scaled Laguerre functions, defined as follows:

• GL nodes

  $$\begin{aligned} \hat{d}^\beta _{ij} = {\left\{ \begin{array}{ll} \dfrac{\hat{{\mathscr {L}}}_q^\beta (z_i^\beta )}{(z_i^\beta -z_j^\beta )\hat{{\mathscr {L}}}_q^\beta (z_j^\beta )} &{} i\ne j \\ \\ -\dfrac{q+2}{2z_i^\beta } &{} i=j \\ \end{array}\right. } \end{aligned}$$

  (74)

• GLR nodes

  $$\begin{aligned} \hat{d}^\beta _{ij} = {\left\{ \begin{array}{ll} \dfrac{\hat{{\mathscr {L}}}_{q+1}^\beta (z_i^\beta )}{(z_i^\beta -z_j^\beta )\hat{{\mathscr {L}}}_{q+1}^\beta (z_j^\beta )} &{} i\ne j \\ \\ 0 &{} i=j\ne 0 \\ \\ -\beta \dfrac{q+1}{2} &{} i=j=0. \\ \end{array}\right. } \end{aligned}$$

  (75)

We also define as $\hat{\varvec{\varOmega }}_\beta$ the diagonal matrix with the quadrature weights ${\hat{\omega }}_i^\beta$ on the diagonal. For a nodal discretization based on Laguerre polynomials, instead, the differentiation matrix ${\mathbf {D}}_\beta$ has entries $d^\beta _{ij}$ defined as:

• GL nodes

  $$\begin{aligned} {d}_{ij}^\beta = {\left\{ \begin{array}{ll} \dfrac{{{\mathscr {L}}}_q^\beta (z_i^\beta )}{(z_i^\beta -z_j^\beta ){{\mathscr {L}}}_q^\beta (z_j^\beta )} &{} i\ne j \\ \\ \dfrac{\beta z_i^\beta -q-2}{2z_i^\beta } &{} i=j \\ \end{array}\right. } \end{aligned}$$

  (76)

• GLR nodes

  $$\begin{aligned} {d}_{ij}^\beta = {\left\{ \begin{array}{ll} \dfrac{{{\mathscr {L}}}_{q+1}^\beta (z_i^\beta )}{(z_i^\beta -z_j^\beta ){{\mathscr {L}}}_{q+1}^\beta (z_j^\beta )} &{} i\ne j \\ \\ \dfrac{\beta }{2} &{} i=j\ne 0 \\ \\ -\beta \dfrac{q}{2} &{} i=j=0 \\ \end{array}\right. } \end{aligned}$$

  (77)

We also set

$$\begin{aligned} \hat{{\mathbf {g}}_1}= & {} [(\hat{h}_0^\beta )^{\prime \prime }(z_1),\dots ,(\hat{h}_0^\beta )^{\prime \prime }(z_q)],\;\hat{{\mathbf {g}}_2}=[(\hat{h}_0^\beta )^{\prime }(z_1),\dots ,(\hat{h}_0^\beta )^{\prime }(z_q)],\\ \mathbf {g_1}= & {} [(h_0^\beta )^{\prime \prime }(z_1),\dots ,(h_0^\beta )^{\prime \prime }(z_q)],\;\mathbf {g_2}=[(h_0^\beta )^{\prime }(z_1),\dots ,(h_0^\beta )^{\prime }(z_q)],\\ \hat{{\mathbf {h}}}= & {} [\hat{h}^\beta _0(0),\dots ,\hat{h}^\beta _q(0)],\; {\mathbf {h}}=[h^\beta _0(0),\dots ,h^\beta _q(0)],\\ \mathbf {\widehat{W}}= & {} \hat{\varvec{\varOmega }}_\beta ^{-1}\hat{{\mathbf {D}}}_\beta ^T\hat{\varvec{\varOmega }}_\beta ,\; {\mathbf {W}}=\varvec{\varOmega }_\beta ^{-1}{\mathbf {D}}_\beta ^T\varvec{\varOmega }_\beta ,\\ \hat{{\mathbf {r}}}= & {} \hat{\varvec{\varOmega }}_\beta ^{-1}\hat{{\mathbf {h}}},\; {\mathbf {r}}=\varvec{\varOmega }_\beta ^{-1}{\mathbf {h}},\;{\mathbf {e}}=[1,\dots ,1]^T\in {\mathbf {R}}^{q+1}. \end{aligned}$$

We also denote by $(\hat{{\mathbf {D}}}_\beta )_q$ for scaled Laguerre functions, and by $({\mathbf {D}}_\beta )_q$ for scaled Laguerre polynomials, the matrices obtained from the differentiation matrices $\hat{{\mathbf {D}}}_\beta$ and ${\mathbf {D}}_\beta$ by removing the first row and the first column. Finally we denote by $(\hat{{\mathbf {D}}}_\beta )_0$ for scaled Laguerre functions, and $({\mathbf {D}}_\beta )_0$ for scaled Laguerre polynomials, the matrices obtained from $\hat{{\mathbf {D}}}_\beta$ and ${\mathbf {D}}_\beta$ by replacing the first row with zeros. The expressions of matrix ${\mathbf {A}}$ and right-hand side ${\mathbf {g}}$ for the derived discretizations are summarized in Table 11—note the two use of the matrix $\hat{{\mathbf {L}}}$ (72) for scaled Laguerre functions and ${\mathbf {L}}$ (73) for scaled Laguerre polynomials.

Table 11 Definition of matrix ${\mathbf {A}}$ and vector ${\mathbf {g}}$ in the Laguerre discretization of (39) or (67) for several formulations ‘Form’, basis functions ‘BF’ and boundary conditions ‘BC’. ‘Coll’: collocation, ‘Nod’: nodal, ‘Mod’: modal, ‘Dir’: Dirichlet, ‘Neu’: Neumann, ‘LF’: Scaled Laguerre Functions, ‘LP’: Scaled Laguerre Polynomials. See text for symbol definitions

As customary for the advection–diffusion problem, the stability property can be a function of the Péclet number, which is usually defined as $Pe=u\mathscr {L}/\mu$, where $\mathscr {L}$ is a reference length scale. For simplicity we choose the length scale $\mathscr {L}=1$, set $\mu =1$ and analyze the stability of ${\mathbf {A}}$ for a fixed value of Pe; the corresponding ranges for $\beta$ are shown in Table 12 for both scaled Laguerre functions and polynomials.

Table 12 Stability of ${\mathbf {A}}$ as a function of $\beta$: condition under which the largest real part of the eigenvalues is non-positive

It can be observed that only the strong form discretizations based on Laguerre functions are stable for all boundary conditions and independently of the value of the Péclet number. Other discretizations based on Laguerre functions are instead stable under mild conditions on the value of $\beta$ as a function of the Péclet number. These conditions become problematic only in the very large Péclet number limit.

In this paper, only the weak form modal discretization based on Laguerre functions was considered for the extended DG scheme, due to its hierarchical nature, that allows in principle for an easy (and if necessary, dynamic) adjustment of the number of basis functions to perform $p-$adaptation. The strong form nodal discretization based on Laguerre functions seems otherwise the most robust option and will be further studied as a basis for extended DG approaches in future work. Discretizations based on Laguerre polynomials are instead only stable under more restrictive conditions, which also affect the choice of $\beta$ in the small Péclet number case. These conclusions complement the results in [7], where the pure advection problem was discussed. Such an analysis does not seem to have been carried out in the literature, to the best of the authors’ knowledge.