Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity

We propose two tailored finite point methods for the advection–diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uniform convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.

Authors and Affiliations

1. Department of Mathematics and Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

   Min Tang

2. Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

   Yihong Wang

Corresponding author

Correspondence to Min Tang.

Min Tang is partially supported by NSFC 11301336 and 91330203.

Appendix

Here, we show that the limiting scheme for the degenerate equation is consist and explain heuristically the monotonicity. We only consider the case when \(\beta _1=\beta _2=0\) and \(\theta \ne 0\). In the limit that \(\epsilon _{i,j}\rightarrow 0\) in (25),

Let the coefficient matrix \(D_k\) in (34) be \((D_k^1,~D_k^2),~k=1,2,3,4\), where \(D_k^i\) (\(i=1,2\)) are \(3\times 1\) columns. We need to show that (34) becomes a consistent and monotone discretization for

Due to the form of \(\gamma _1\), \(\gamma _2\), \(\gamma _3\), (49) can be rewritten as

If we are able to show that when \(\epsilon _{i,j}\rightarrow 0\), the limit of (34) is a discretization to (50), the proof completes.

For the degenerate case we consider here, \(\beta _{1,i,j}=\beta _{2,i,j}=\epsilon _{i,j}=0\), so that \((\lambda _{i,j}, \mu _{i,j})\) in (31) become \((\pm \sqrt{\frac{1}{\alpha \cos ^2\theta }},0)\), \((0,\pm \sqrt{\frac{1}{\alpha \sin ^2\theta }})\). Let

the linear space \(H_{i,j}\) becomes

When \((\lambda _{k,i,j},\mu _{k,i,j})=(\pm \lambda ,0)\), \((\lambda _{k,i,j},\mu _{k,i,j})=(0,\pm \mu )\), (35) are respectively

and (36) are respectively (after dividing some constant)

Let

The subtraction of the two equations in (52), the subtraction of the two equations in (53) and the summation of the two equations in (54), the summation of the two equations in (54) yield a linear system with scalar coefficnets for

Since the right hand side of the subsystem for \(D_{1,1}^1-D_{2,1}^1\), \(D_{3,1}^1-D_{4,1}^1\), \(D_{1,1}^2+D_{2,1}^2\) and \(D_{3,1}^3+D_{4,1}^2\) is zero, we get

The summation of the two equations in (52), the summation of the two equations in (53) and the subtraction of the two equations in (54), the subtraction of the two equations in (54) yield a linear system with scalar coefficients for

which provides the following system for \(g_1\), \(g_2\), \(g_3\), \(g_4\)

The first line of the scheme (34) is

The system (57) can be solved exactly such that

Then by Taylor expansion, when \(\theta \) is away from \(n\pi /2\) (\(n=0,1,2,3\)) and the mesh is fine enough,

• The second order convergence. Let

  $$\begin{aligned} g=\frac{\cosh (\lambda h_x)\cos (\mu h_y)-1}{(\cosh (\lambda h_x)-1)(\cosh (\mu h_y)-1)} \end{aligned}$$

  Multiplying both sides of (58) by g and using (51), (59), we can obtain

  $$\begin{aligned} \begin{aligned}&\displaystyle -\alpha \cos ^2\theta \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h_x^2}-\alpha \sin ^2\theta \frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h_y^2}\\&\displaystyle -2\alpha \sin ^3\theta \cos \theta \frac{\partial _y u_{i+1,j}-\partial _y u_{i-1,j}}{2h_x}-2\alpha \cos ^3\theta \sin \theta \frac{\partial _x u_{i,j+1}-\partial _x u_{i,j-1}}{2h_y}\\&+u_{i,j}=O(h_x^2+h_y^2). \end{aligned} \end{aligned}$$

  (60)

  Therefore the limiting scheme is a second order discretization for (50).

• Heuristic argument about the monotonicity. When \(\theta \) is away from \(n\pi /2\) with \(n=0,1,2,3\), i.e. \(\sin \theta ,\cos \theta \ne 0\), let

  $$\begin{aligned} \begin{aligned} \digamma _1&= u_{i+1,j}+\frac{\sin \theta }{\cos \theta }h_x\partial _y u_{i+1,j}-2u_{i,j}+u_{i-1,j}-\frac{\sin \theta }{\cos \theta }h_x\partial _y u_{i-1,j},\\ \digamma _2&= u_{i,j+1}+\frac{\cos \theta }{\sin \theta }h_y\partial _x u_{i,j+1}-2u_{i,j}+u_{i,j-1}-\frac{\cos \theta }{\sin \theta }h_y\partial _x u_{i,j-1},\\ \digamma _3&=\left( g_1-\frac{g_3}{\frac{\sin \theta }{\cos \theta }h_x}\right) \left( u_{i+1,j}+u_{i-1,j}\right) +\left( g_2-\frac{g_4}{\frac{\cos \theta }{\sin \theta }h_y}\right) (u_{i,j+1}+u_{i,j-1})-u_{i,j}. \end{aligned} \end{aligned}$$

  (61)

  We can reformulate (58) into

  $$\begin{aligned} \displaystyle \frac{g_3}{\frac{\sin \theta }{\cos \theta }h_x}\digamma _1+\frac{g_4}{\frac{\cos \theta }{\sin \theta }h_y} \digamma _2+\digamma _3+\left( 2\frac{g_3}{\frac{\sin \theta }{\cos \theta }h_x}+2\frac{g_4}{\frac{\cos \theta }{\sin \theta }h_y}\right) u_{i,j}=0. \end{aligned}$$

  (62)

  According to the definition of monotone schemes for degenerate elliptic equation in [16], in order to have a consist and convergent scheme for the degenerate elliptic equation, we have to show that the coefficients in front of all neighboring grids are positive while the coefficient in front of \(u_{i,j}\) are negative. For simplicity we only consider the case when \(\theta \in (0,\pi /2)\). First of all, \(g_1,g_2,g_3,g_4>0\) in this case.

  $$\begin{aligned} \begin{aligned} u_{i+1,j}+\frac{\sin \theta }{\cos \theta }h_x\partial _y u_{i+1,j},\qquad&u_{i-1,j}-\frac{\sin \theta }{\cos \theta }h_x\partial _y u_{i-1,j},\\ u_{i,j+1}+\frac{\cos \theta }{\sin \theta }h_y\partial _x u_{i,j+1},\qquad&u_{i,j-1}-\frac{\cos \theta }{\sin \theta }h_y\partial _x u_{i,j-1} \end{aligned} \end{aligned}$$

  can be respectively considered as approximations to the function values at \(Z_{i,j}^1\), \(Z_{i,j}^2\), \(Z_{i,j}^3\), \(Z_{i,j}^4\) in Fig. 12. Besides, from (59) and \(g_1,g_2>0\), the coefficients in front of \(u_{i\pm 1,j}\), \(u_{i,j\pm 1}\) in \(\digamma _3\) are positive. Finally, the coefficient in front of \(u_{i,j}\) is \(-1\) which is negative, therefore the limiting scheme is monotone, which gives the convergence of the scheme for the degenerate equation according to [16]. When \(\theta \) is in other quadrant, the discussion is the same.

  The approximations in (59) can only be valid the mesh have to be fine enough, especially when \(\theta \) is really close to \(n\pi /2\). One may argue that when \(\theta \) is close to \(n\pi /2\) with \(n=0,1,2,3\), the locations of \(Z_{i,j}^1\), \(Z_{i,j}^2\), \(Z_{i,j}^3\), \(Z_{i,j}^4\) can be far away from the cell. This is not a proof but a heuristic argument. However, the numerical results seem quite promising.

The limiting scheme

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