TimeDependentEquationSystem

For transient problems, time derivatives $var element = document.getElementById("moose-equation-9284cfaf-a216-44cb-acf8-0bb68f7ea825");katex.render("\\dot{u}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ of the trial variables $var element = document.getElementById("moose-equation-1630a780-d59c-4944-a0fa-a5f6d7ce7dd9");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ will also be present in the weak form.

var element = document.getElementById("moose-equation-221d1267-f83b-4ce3-92a6-1ee66326583d");katex.render("\\left(\\mathcal{T}(\\dot{u}), v\\right)_{\\Omega} + {\\left(\\mathcal{L}(u), v\\right)_{\\Omega} =\\left(f,v\\right)_{\\Omega}\\,\\,\\,\\forall v \\in V}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Contributions to $var element = document.getElementById("moose-equation-7eb7b96e-6200-4a23-8ae0-832baf5a8401");katex.render("\\left(\\mathcal{T}(\\phi_j), \\varphi_i\\right)_{\\Omega}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are given by time derivative kernels such as MFEMTimeDerivativeMassKernel.

Currently, transient problems in MOOSE's wrapping of MFEM are solved using an implicit backwards Euler method. Denoting $var element = document.getElementById("moose-equation-45abf7d5-ba7f-4ec0-afe0-d31046db8f20");katex.render("u_n=u(t+\\delta t)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-83b2a6a4-ac6f-4d4d-97b2-80ede7b7c71e");katex.render("u_{n-1}=u(t)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and approximating $var element = document.getElementById("moose-equation-07978e88-d413-41cf-9e58-81b767f54d4f");katex.render("u(t+\\delta t) \\approx u(t) + \\delta t \\dot{u}(t+\\delta t)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , we have

var element = document.getElementById("moose-equation-6684d14d-5bdf-4c4c-ae17-87926c85caa2");katex.render("\\left([\\mathcal{T}+\\delta t\\mathcal{L}](u_n), v\\right)_{\\Omega} =\\left([\\delta t f + \\mathcal{T}(u_{n-1})],v\\right)_{\\Omega}\\,\\,\\,\\forall v \\in V", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});