Linearization of viscous flux in discontinuous Galerkin finite element methods

yidongxia · November 17, 2011, 11:28

I'm a student studying discontinuous Galerkin (DG) finite element methods in CFD.

Many papers related to implicit DG methods for Navier-Stokes equations claim that the linearization of viscous flux (viscous Jacobian matrix) is computed analytically, but provide no more details.

In finite volume methods, it's not necessary to compute the viscous Jacobian matrix explicitly. A common simplification is to use the spectral radius instead of analytically derived viscous Jacobian matrix. However for DG methods, I haven't seen any papers addressing the derivation of viscous Jacobians explicitly. So far as my own experience can tell, if the linearization of viscous flux is not treated carefully, the implicit solver will not work.

If it happens that you are familiar with DG methods, I'll appreciate so much if you would share your experience and discuss with me

praveen · November 17, 2011, 12:11

If you treat any coefficients in the viscous terms as constants, then the viscous terms are linear, e.g., $\frac{d}{dx}\mu \frac{du}{dx}$ , consider $\mu$ to be constant for derivation of implicit scheme. Assembling such terms will give you A*U where A is a matrix and U is the vector of unknown solution. The linearization of this is just A*dU. At the analytical level, you have a bilinear for a(u,v) representing the viscous terms. Its linearlization is just a(du, v) whose assembled version is A*dU.

November 17, 2011, 11:28	Linearization of viscous flux in discontinuous Galerkin finite element methods	#1
yidongxia New Member Yidong Join Date: Nov 2011 Posts: 23 Rep Power: 14	I'm a student studying discontinuous Galerkin (DG) finite element methods in CFD. Many papers related to implicit DG methods for Navier-Stokes equations claim that the linearization of viscous flux (viscous Jacobian matrix) is computed analytically, but provide no more details. In finite volume methods, it's not necessary to compute the viscous Jacobian matrix explicitly. A common simplification is to use the spectral radius instead of analytically derived viscous Jacobian matrix. However for DG methods, I haven't seen any papers addressing the derivation of viscous Jacobians explicitly. So far as my own experience can tell, if the linearization of viscous flux is not treated carefully, the implicit solver will not work. If it happens that you are familiar with DG methods, I'll appreciate so much if you would share your experience and discuss with me

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November 17, 2011, 12:11		#2
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 342 Blog Entries: 6 Rep Power: 18	If you treat any coefficients in the viscous terms as constants, then the viscous terms are linear, e.g., $\frac{d}{dx}\mu \frac{du}{dx}$ , consider $\mu$ to be constant for derivation of implicit scheme. Assembling such terms will give you AU where A is a matrix and U is the vector of unknown solution. The linearization of this is just AdU. At the analytical level, you have a bilinear for a(u,v) representing the viscous terms. Its linearlization is just a(du, v) whose assembled version is A*dU.