From 884b78dc2915fd703f8ef83ae9b996ba35847986 Mon Sep 17 00:00:00 2001 From: "Daniel R. Reynolds" Date: Fri, 4 Oct 2024 11:06:01 -0500 Subject: [PATCH] Added comment regarding order of MERK54 method --- .../guide/source/Usage/MRIStep/MRIStepCoupling.rst | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/doc/arkode/guide/source/Usage/MRIStep/MRIStepCoupling.rst b/doc/arkode/guide/source/Usage/MRIStep/MRIStepCoupling.rst index 0e3b1ca570..885e7c8ca1 100644 --- a/doc/arkode/guide/source/Usage/MRIStep/MRIStepCoupling.rst +++ b/doc/arkode/guide/source/Usage/MRIStep/MRIStepCoupling.rst @@ -399,6 +399,15 @@ with values specified for each method below (e.g., ``ARKODE_MIS_KW3``). for each order when using adaptive time stepping is marked with a circle (:math:`^\circ`). The "Slow RHS Calls" column corresponds to the number of calls to the slow right-hand side function, :math:`f^E`, per time step. + Note :math:`^\vee`: although all MERK methods were derived in :cite:p:`Luan:20` + under an assumption that the fast time scale is linear in the solution, in + :cite:p:`Fish:24` it was proven that they also satisfy all nonlinear order + conditions up through their linear order. The lone exception is MERK54, where + it was only proven to satisfy all nonlinear conditions up to order 4, since + :cite:p:`Fish:24` did not establish the formulas for the order 5 conditions. + All our numerical tests to date have shown MERK54 to achieve fifth order for + nonlinear problems, and so we conjecture that it also satisfies the nonlinear + fifth order conditions. ====================================== ================== =============== ============== ===================== Table name Method Order Embedding Order Slow RHS Calls Reference @@ -414,7 +423,7 @@ with values specified for each method below (e.g., ``ARKODE_MIS_KW3``). :index:`ARKODE_MERK32` 3 2 3 :cite:p:`Luan:20` :index:`ARKODE_MRI_GARK_ERK45a` :math:`4^{*\circ}` 3 5 :cite:p:`Sandu:19` :index:`ARKODE_MERK43` 4 3 6 :cite:p:`Luan:20` - :index:`ARKODE_MERK54` 5 4 10 :cite:p:`Luan:20` + :index:`ARKODE_MERK54` :math:`5^{\vee}` 4 10 :cite:p:`Luan:20` ====================================== ================== =============== ============== =====================