Multi-dimensional schemes for scalar advection

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77204/1/AIAA-1991-1532-834.pd

Progress on multidimensional upwind Euler solvers for unstructured grids

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76503/1/AIAA-1991-1550-511.pd

A multidimensional generalization of Roe's flux difference splitter for the euler equations

Upwind methods for the 1-D Euler equations, such as TVD schemes based on Roe's approximate Riemann solver, are reinterpreted as residual distribution schemes, assuming continuous piecewise linear space variation of the unknowns defined at the cell vertices. From this analysis three distinct steps are identified, each allowing for a multidimensional generalization without reference to dimensional splitting or 1-D Riemann problems. A key element is the necessity to have continuous piecewise linear variation of the unknowns, requiring linear triangles in two space dimensions and tetrahedra in three space dimensions. Flux differences naturally generalize to flux contour integrals over the triangles. Roe's flux difference splitter naturally generalizes to a multidimensional flux balance splitter if one assumes that the parameter vector variable is the primary dependent unknown having linear variation in space. Nonlinear positive and second-order scalar distribution schemes provide a true generalization of the TVD schemes in one space dimension. Although refinements and improvements are still possible for all these elements, computational examples show that these generalizations present a new framework for solving the multidimensional Euler equations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30942/1/0000612.pd

Adjoint error estimation for residual based discretizations of hyperbolic conservation laws I : linear problems

The current work concerns the study and the implementation of a modern algorithm for error estimation in CFD computations. This estimate involves the dealing of the adjoint argument. By solving the adjoint problem, it is possible to obtain important information about the transport of the error towards the quantity of interest. The aim is to apply for the first time this procedure into Petrov-Galerkin (PG) method. Streamline Upwind Petrov-Galerkin, stabilised Residual Distribution and bubble method are involved for the implementation. Scalar hyperbolic problems are firstly used as test cases

Computations of inviscid compressible flows using fluctuation-splitting on triangular meshes

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77018/1/AIAA-1993-3301-532.pd

Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws

A class of truly multidimensional upwind schemes for the computation of inviscid compressible flows is presented here, applicable to unstructured cell-vertex grids. These methods use very compact stencils and produce sharp resolution of discontinuities with no overshoots.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47813/1/466_2004_Article_BF00350091.pd

Characterisation of magnesium ascorbyl phosphate, a raw material in cell therapy

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A second order unconditionally positive space-time residual distribution method for solving compressible flows on moving meshes

summary:A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$, $n+1/2$ and $n+1$. The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level $n+1/2$ with a limited variant of a positive first order scheme, ensuring monotonicity and second order of accuracy in smooth flow under a time-step restriction for the timestep of the first layer. The space-time residual from the upper layer of the elements is distributed to both levels $n+1/2$ and $n+1$, with a similar scheme, giving monotonicity without any time-step restriction. The two-layer scheme allows a time marching procedure thanks to initial value condition imposed on the first layer of elements. The scheme is positive and second order accurate in space and time for arbitrary meshes and it satisfies the Geometric Conservation Law condition (GCL) by construction. Example calculations are shown for the Euler equations of inviscid gas dynamics, including the 1D problem of gas compression under a moving piston and transonic flow around an oscillating NACA0012 airfoil