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Discontinuous Galerkin Methods for Solving Euler Equations Mid Year Presentation/Status Update December 6th, 2012 Andrey Andreyev ([email protected]) Advisor: James Baeder ([email protected]) Objectives: Develop a Discontinuous Galerkin Method to solve the Euler Equations in one dimension that allows for up to 3rd spatial order discretization Develop a Discontinuous Galerkin Method to Solve the Euler Equation in two dimensions that allows for up to 3rd order spatial discretization Validate both codes against known solutions. Shock tube problem for the one dimensional case and a known airfoil section for the two dimensional case The Euler Equations General Form One Dimensional Form U j F(U) 0 t U ui E U F(U) 0 t x U u E u j F(U) ui u j ij p u ( E p) j u 2 F(U) u p u ( E p ) Spatial Discretization Structured Mesh Unstructured Mesh Picture from: http://www.cglerlangen.com/downloads/Manual/ch09s16s01.html Picture from: http://ta.twi.tudelft.nl/users/wesselin/projects/un structured.html More on spatial discretization and accuracy Picture of stencil 1st and 2ndorder first derivative approximation u j x u j x Figure From Computational Gasdynamics: Laney2 u j 1 u j x j 1 x j u j 1 u j 1 2( x j 1 x j 1 ) Discretization, Conservation and Flux Capturing Require scheme to capture shocks and other discontinuities “automatically” and not using “shock fitting methods” Higher spatial order shock capturing schemes (>2nd order) tend to be more oscillatory around the discontinuities because of the larger stencils required thus more points are contributing around areas with large gradients Figure from Computational Gasdynamics: Laney 2 Overview of Current Computational Approaches In general, methods in Computational Fluid Dynamics can be divided into three approaches: Finite Difference Methods Advantages: • Ease of Implementation • Easy to make higher order Disadvantages: • Only applicable on structured grids Finite Element Advantages: • Can be any order of accuracy • Based on variational methods • Applicable on unstructured grids Disadvantages: • More complex • Not conservative! • Naturally implicit (can be explicit with modifications) Finite Volume Advantages: • Naturally Conservative (captures discontinuities in the flow field) • Many upwinding possibilities • Applicable on unstructured grids Disadvantages: • Difficult to devise stable higher order scheme General Discontinuous Galerkin Setup 1. Start with the Euler Equation: U F(U) 0 t 2. Discretize the spatial domain and assume and assume an approximate solution on a perelement basis U uh u j (t )vj 3. Multiply by weight function and integrate by parts vk ui , j (t )vjd vk Fid vkFiR dS 0 t Note the boundary term has a different flux term. In normal finite element, the boundary terms need to enforce connectivity with neighboring elements. In Discontinuous Galerkin Methods the boundary fluxes are calculated using the Riemann Fluxes. This enforces connectivity and allows for discontinuities in the solution! One-Dimensional Discontinuous Galerkin1 U F(U) 0 t x U u E u F(U) u 2 p u ( E p ) Require an approximation to the solution in the form of: k u al u (jl ) (t )vl( j ) ( x)dx h j al l 1 Define the shape function as: v1j 1 v2j x x j v2j ( x x j ) 2 x lj ( j) 2 [ v ( x )] dx l Ij 1 x 2j 12 Define the degrees of freedom as: 1 u(jl ) (t ) l u( x, t )vl( j ) dx x I j Note: 1st DOF is the cell average of the conservative variables In Galerkin method the weight functions are taken to be the same as the shape functions. h Multiplying the Euler Equations by the weight functions and substituting u for U and integrating by parts, we obtain the following form: d 1 u (jl ) [vl( j ) ( x j 1 / 2 )f (u j 1 / 2 ) vl( j ) ( x j 1/ 2 )f (u j 1/ 2 )] l dt x j 1 x lj h F ( u ( x, t ) I j d vl( j ) d x 0 dx Shape Functions over each element Exact solution to the Riemann Problem: Interface fluxes2 The second term of in the last equation has not been defined yet. How do we get the fluxes at the cell interfaces? The Riemann Problem has an exact solution! Consider an Euler Equation with the initial of: u u ( x, t o ) L u R x xo x xo Expansion Fan: Figure from Computational Gasdynamics: Laney 2 Consider that every cell interface is a Riemann problem! Exact solution to Riemann problem is very expensive and we are not interested in in the solution at all x/t. Look for a suitable approximation for x/t=0 only via Roe Averages 1 r1 u RL 1 2 2 u RL r2 1 u RL aRL h a u RL RL RL RL 2aRL r2 All equations taken from Laney2 u RL aRL h a u RL RL RL 2aRL 1 Slope Limiting for Stability1 Around discontinuities DOFs representing the gradients are very large causing oscillations and instabilities. To remedy this problem slope limiters are introduced to insure stability Runge-Kutta Time Explicit Time Marching •Time integration of the equations will be carried out using a higher order Runge-Kutta technique. •The space discretization in the previous slide converted the PDEs into a system of ODEs in time. • Using Higher Order Runge-Kutta, we carry out the time integration on a perelement basis t CFL min u j a j x j Note: Time Step is calculated based on the largest Eigenvalue i.e. fastest information transfer DG Method Test Problems The method will first be implemented on the one-dimensional version of the Euler Equations to test the methods accuracy. Sod’s Shock Tube Problem will be used as the test case since it has an exact solution and will test the scheme’s shock capturing ability. Implemented using Fortran 95 Exact Solution Image: Author Generated Image: http://en.wikipedia.org/wiki/File:SodShockTubeTest_ Regions.png 1st Order Spatial Discretization Results Density Evolution Mach Number Evolution 2nd Order Spatial Discretization Results Density Evolution Mach Number Evolution 3rd Order Spatial Discretization Results Density Evolution Mach Number DG Method Test Problems Two Dimensional Airfoil on a structured mesh. Mesh provided by Dr. Baeder. Boundary Conditions: Tangential Flow around the airfoil (Inviscid Wall) Undisturbed flow at the domain edge (Farfield) Run at a variety of mach numbers to create different flow regimes (subsonic, trans-sonic, supersonic) to steady state Compare the results to experimental data (large database for many airfoils) Compare results to other established computational tools Implemented serially using Fortran 95, then parallelized using MPI http://www.salomeplatform.org/forum/forum_10/213329959/view DG Method Implementation Original Schedule 10/31/12- 12/15/1302/15/1303/15/13- One dimensional version. Apply to one dimensional problem with a known solution to test accuracy and shock capturing abilities. Sod shock tube problem. Will validate the 1-D version (serial) Two dimensional version. Apply to 2-D airfoil problem using provided grids (serial) Validation of the two dimensional version using experimental airfoil results as well as the results published in literature Parallelization of the two dimensional. Validate using results from the serial version Questions??? References: 1. Bernardo Cockburn, Chi-Wang Shu. TVB Runge-Kutta Local Projection Discontinuous Galerkin Method for Conservation Laws II: General Frame Work. Mathematics of Computation Volume 52, Issue 186, yr. 1989 2. Culbert B. Laney. Computational Gasdynamics. Cambridge University Press. 1998.