From d446c471a0da882c40441e27b572863d9c6f58aa Mon Sep 17 00:00:00 2001
From: Timo Koch <timokoch@uio.no>
Date: Mon, 3 Jul 2023 11:56:45 +0200
Subject: [PATCH] [multistage][doc] Improve documentation
---
.../timestepping/multistagemethods.hh | 17 +++++++++++++----
1 file changed, 13 insertions(+), 4 deletions(-)
diff --git a/dumux/experimental/timestepping/multistagemethods.hh b/dumux/experimental/timestepping/multistagemethods.hh
index 3533d7f7b2..3d000912a6 100644
--- a/dumux/experimental/timestepping/multistagemethods.hh
+++ b/dumux/experimental/timestepping/multistagemethods.hh
@@ -39,12 +39,12 @@ namespace Dumux::Experimental {
*
* \f[
* \begin{equation}
- * \frac{\partial M(x)}{\partial t} - R(x, t) = 0,
+ * \frac{\partial M(x, t)}{\partial t} - R(x, t) = 0,
* \end{equation}
* \f]
*
- * where \f$ M(x)\f$ is the temporal operator/residual in terms of the solution \f$ x \f$,
- * and \f$ R(x)\f$ is the discrete spatial operator/residual.
+ * where \f$ M(x, t)\f$ is the temporal operator/residual in terms of the solution \f$ x \f$,
+ * and \f$ R(x, t)\f$ is the discrete spatial operator/residual.
* \f$ M(x)\f$ usually corresponds to the conserved quantity (e.g. mass), and \f$ R(x)\f$
* contains the rest of the residual. We can then construct \f$ m \f$-stage time discretization methods.
* Integrating from time \f$ t^n\f$ to \f$ t^{n+1}\f$ with time step size \f$ \Delta t^n\f$, we solve:
@@ -53,13 +53,22 @@ namespace Dumux::Experimental {
* \begin{aligned}
* x^{(0)} &= u^n\\
* \sum_{k=0}^i \left[ \alpha_{ik} M\left(x^{(k)}, t^n + d_k\Delta t^n\right)
- * + \beta_{ik}\Delta t^n R \left(x^{(k)}, t^n+d_k\Delta t^n \right)\right] &= 0 & \forall i \in \{1,\ldots,m\} \\
+ * + \beta_{ik}\Delta t^n R \left(x^{(k)}, t^n+d_k\Delta t^n \right)\right]
+ * &= 0 & \forall i \in \{1,\ldots,m\} \\
* x^{n+1} &= x^{(m)}
* \end{aligned}
* \f]
* where \f$ x^{(k)} \f$ denotes the intermediate solution at stage \f$ k \f$.
* Dependent on the number of stages \f$ m \f$, and the coefficients \f$ \alpha, \beta, d\f$,
* schemes with different properties and order of accuracy can be constructed.
+ *
+ * That the summation only goes up to \f$ i \f$ in stage \f$ i \f$ means that we
+ * restrict ourselves to diagonally-implicit Runge-Kutta schemes (DIRK)
+ * and explicit schemes.
+ *
+ * Note that when computing the Jacobian of the residual with respect
+ * to \f$ x^{(k)} \f$ at stage \f$ k \f$, only the terms containing the solution of
+ * the current stage \f$ k \f$ contribute to the derivatives.
*/
template<class Scalar>
class MultiStageMethod
--
GitLab