From f9c43b2029f7e920de128d5295852f0b3a8aab07 Mon Sep 17 00:00:00 2001 From: Kyle Niemeyer Date: Mon, 2 Oct 2023 16:52:01 -0700 Subject: [PATCH] minor edits in 2nd-order backward euler --- book/content/second-order/numerical-methods.ipynb | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/book/content/second-order/numerical-methods.ipynb b/book/content/second-order/numerical-methods.ipynb index 5e0f23a..fab19b2 100644 --- a/book/content/second-order/numerical-methods.ipynb +++ b/book/content/second-order/numerical-methods.ipynb @@ -338,10 +338,11 @@ "\\begin{align}\n", "y_{1, i+1} &= y_{1, i} + \\Delta t \\, y_{2,i+1} \\\\\n", "y_{2, i+1} &= y_{2, i} + \\Delta t \\left( 10 - 6 y_{2, i+1} - 5 y_{1,i+1} \\right) \\\\\n", - "\\\\\n", - "y_{1, i+1} - \\Delta t \\, y_{2, i+1} &= y_{1,i} \\\\\n", - "5 \\Delta t \\, y_{1, i+1} + (1 + 6 \\Delta t) y_{2, i+1} &= y_{2,i} + 10 \\Delta t \\\\\n", - "\\text{or} \\quad \n", + "\\rightarrow y_{1, i+1} - \\Delta t \\, y_{2, i+1} &= y_{1,i} \\\\\n", + "5 \\Delta t \\, y_{1, i+1} + (1 + 6 \\Delta t) y_{2, i+1} &= y_{2,i} + 10 \\Delta t\n", + "\\end{align}\n", + "or, represented as a product of a coefficient matrix and a vector:\n", + "\\begin{align}\n", "\\begin{bmatrix} 1 & -\\Delta t \\\\ 5 \\Delta t & (1+6\\Delta t)\\end{bmatrix} \n", "\\begin{bmatrix} y_{1, i+1} \\\\ y_{2, i+1} \\end{bmatrix} &= \n", "\\begin{bmatrix} y_{1,i} \\\\ y_{2,i} + 10 \\Delta t \\\\ \\end{bmatrix} \\\\\n",