ℒ-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations

Abstract

Well-balanced schemes, nowadays mostly developed for both hyperbolic and kinetic equations, are extended in order to handle linear parabolic equations, too. By considering the variational solution of the resulting stationary boundary-value problem, a simple criterion of uniqueness is singled out: the $C^{1}$ regularity at all knots of the computational grid. Being easy to convert into a finite-difference scheme, a well-balanced discretization is deduced by defining the discrete time-derivative as the defect of $C^{1}$ regularity at each node. This meets with schemes formerly introduced in the literature relying on so-called ℒ-spline interpolation of discrete values. Various monotonicity, consistency and asymptotic-preserving properties are established, especially in the under-resolved vanishing viscosity limit. Practical experiments illustrate the outcome of such numerical methods.

Appendix: An External Force \(f(x)\) in a Reaction-Diffusion Equation

Hereafter, the aim is to solve the initial-boundary problem,

$$ \partial_{t}u -\varepsilon \partial_{xx}u + q(x)u = f(x), \quad u(t, \pm 1)=u^{\pm }, \quad x \in (-1,1), $$

(A.1)

so that the resulting scheme recovers exactly the collection of points $v(x_{j})$, for $x_{j} \in \Delta$ the nodes of the grid. So, at each time-step $t^{n}$, the “ℒ-spline” interpolation of the data $u_{j}^{n}$ proceeds by solving the stationary equation between $u_{j}^{n}$ and $u_{j+1}^{n}$. Since, in $(x_{j},x_{j+1})$, both $\bar{q}$ and $\bar{f}$ are constant, each “local profile” is obtained by a standard “variation of constants” technique involving exponential functions:

• let $r_{j-\frac{1}{2}}=\sqrt{\bar{q}/\varepsilon }>0$ in $(x_{j-1},x _{j})$ and $M_{j-\frac{1}{2}}$ be the matrix in,

  $$ \left ( \textstyle\begin{array}{c} u_{j-1}^{n} - \frac{\bar{f}}{\bar{q}} \\ u_{j}^{n} - \frac{\bar{f}}{\bar{q}} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{c@{\quad}c} \exp (r_{j-\frac{1}{2}} x_{j-1}) & \exp (-r_{j-\frac{1}{2}} x_{j-1}) \\ \exp (r_{j-\frac{1}{2}} x_{j}) & \exp (-r_{j-\frac{1}{2}} x_{j}) \\ \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{c} A \\ B \end{array}\displaystyle \right ) . $$

  Yet the determinant $|M_{j-\frac{1}{2}}|= -2\sinh (r_{j-\frac{1}{2}} \Delta x)\neq0$.

• By mimicking Corollary 1, inside $(x_{j-1},x_{j})$,

  $$ v'(x)=r_{j-\frac{1}{2}} \bigl(A \exp (r_{j-\frac{1}{2}} x) - B \exp (-r _{j-\frac{1}{2}} x) \bigr), $$

  and since $M_{j\pm \frac{1}{2}}$ are invertible, $C^{1}$ smoothness is just:

  $$\begin{aligned} &r_{j-\frac{1}{2}}\Biggl\langle \left ( \textstyle\begin{array}{c} u_{j-1}^{n} - \frac{\bar{f}}{\bar{q}} \\ u_{j}^{n} - \frac{\bar{f}}{\bar{q}} \end{array}\displaystyle \right ) , M_{j- \frac{1}{2}}^{-T}\left ( \textstyle\begin{array}{c} \exp (r_{j-\frac{1}{2}} x_{j}) \\ -\exp (-r_{j-\frac{1}{2}} x_{j}) \end{array}\displaystyle \right ) \Biggr\rangle \\ &\quad =r_{j+\frac{1}{2}}\Biggl\langle \left ( \textstyle\begin{array}{c} u_{j}^{n} - \frac{\bar{f}}{\bar{q}} \\ u_{j+1}^{n} - \frac{\bar{f}}{\bar{q}} \end{array}\displaystyle \right ) , M_{j+ \frac{1}{2}}^{-T}\left ( \textstyle\begin{array}{c} \exp (r_{j+\frac{1}{2}} x_{j}) \\ -\exp (-r_{j+\frac{1}{2}} x_{j}) \end{array}\displaystyle \right ) \Biggr\rangle , \end{aligned}$$

  (A.2)

  where we have used the adjoint of the inverse matrices $M_{j\pm \frac{1}{2}}^{-1}$ so that we don’t need the values of $A,B$. Of course, the $( \frac{\bar{f}}{\bar{q}} ) _{j \pm \frac{1}{2}}$ are different on both sides of the equality. By developing (A.2),

  $$\begin{aligned} &\frac{u_{j}^{n+1}-u_{j}^{n}}{\Delta t}-\frac{\varepsilon }{\Delta x} \biggl[ \frac{r_{j+\frac{1}{2}}}{\sinh (r_{j+\frac{1}{2}}\Delta x)} \bigl( u_{j+1}^{n}-\cosh (r_{j+\frac{1}{2}}\Delta x)u_{j}^{n} \bigr) \\ &\quad {}-\frac{r_{j-\frac{1}{2}}}{\sinh (r_{j-\frac{1}{2}}\Delta x)} \bigl( \cosh (r _{j-\frac{1}{2}}\Delta x)u_{j}^{n}-u_{j-1}^{n} \bigr) \\ &\quad {}+r_{j+\frac{1}{2}}\frac{\cosh (r_{j+\frac{1}{2}}\Delta x)-1}{\sinh (r _{j+\frac{1}{2}}\Delta x)} \cdot \biggl( \frac{\bar{f}}{\bar{q}} \biggr) _{j + \frac{1}{2}} \\ &\quad {} +r_{j-\frac{1}{2}}\frac{\cosh (r_{j-\frac{1}{2}}\Delta x)-1}{ \sinh (r_{j-\frac{1}{2}}\Delta x)}\cdot \biggl( \frac{\bar{f}}{\bar{q}} \biggr) _{j - \frac{1}{2}} \biggr] =0. \end{aligned}$$

  Consistency can be established proceeding like in Proposition 2.