A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs
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We devised a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method for a group of parameterized convection diffusion PDEs with different initial conditions, body forces, boundary conditions and coefficients in our earlier work [3]. We obtained an optimal convergence rate for the ensemble solutions in L^∞(0,T;L^2(Ω)) on a simplex mesh; and obtained a superconvergent rate for the ensemble solutions in L^2(0,T;L^2(Ω)) after an element-by-element postprocessing if polynomials degree k> 1 and the coefficients of the PDEs are independent of time. In this work, we propose a new second order time stepping ensemble HDG method to revisit the problem. We obtain a superconvergent rate for the ensemble solutions in L^∞(0,T;L^2(Ω)) without an element-by-element postprocessing for all polynomials degree k> 0. Furthermore, our mesh can be any polyhedron, no need to be simplex; and the coefficients of the PDEs can dependent on time. Finally, we present numerical experiments to confirm our theoretical results.
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