Per Lötstedt

Implicit solution of hyperbolic equations with space-time adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.

Adaptive Error Control for Steady State Solutions of Inviscid Flow

The steady state solution of the Euler equations of inviscid flow is computed by an adaptive method. The grid is structured and is refined and coarsened in predefined blocks. The equations are discretized by a finite volume method. Error equations, satisfied by the solution errors, are derived with the discretization error as the driving right-hand side. An algorithm based on the error equations is developed for errors propagated along streamlines. Numerical examples from two-dimensional compressible and incompressible flow illustrate the method.

Grid independent convergence of the multigrid method for first-order equations

The multigrid method for numerical solution of systems of first-order partial differential equations is analyzed. The smoothing iterations are of polynomial type and are used on all grids. Under certain conditions the convergence of the iterations is independent of the grid size. This is explained by the propagation of smooth error modes out through the boundary and the damping of oscillatory modes. An upper bound is derived on the number of multigrid V -cycles needed to solve a two-dimensional, scalar equation. Numerical experiments with a simple scalar equation and the Euler equations confirm the theoretical results.

Space---Time Adaptive Solution of First Order PDES

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge–Kutta–Fehlberg (RKF) method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers’ equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.

Accuracy of a propeller model in inviscid flow

A model for computation of the time-averaged inviscid flow around airplane configurations with propellers is described. The propeller is replaced by an actuator disk in the Euler equations. The propeller forces are determined by a combined momentum-blade element theory. The computed results are compared to onedimensional theory, calculations with a panel method, and wind-tunnel experiments. The tested configurations include a full aircraft at subsonic speed. The influence of the grid resolution and numerical parameters is also investigated.

Analysis of multigrid methods for general systems of PDE

Most iteration methods for solving boundary value problems can be viewed as approximations of a time-dependent differential equation. In this paper we show that the multigrid method has the effect of increasing the time-step for the smooth part of the solution leading back to an increase of the convergence rate. For the non-smooth part the convergence is an effect of damping. Fourier analysis is used to find the relation between the convergence rate for multigrid methods and single grid methods. The analysis is performed for general partial differential equations and an arbitrary number of grids.

Parallel single grid and multigrid solution of industrial compressible flow problems

Abstract The Euler and Navier-Stokes equations with a k -ϵ turbulence model are solved numerically in parallel on a distributed memory machine IBM SP2, a shared memory machine SGI Power Challenge, and a cluster of SGI workstations. The grid is partitioned into blocks and the steady state solution is computed using single grid and multigrid iteration. The multigrid algorithm is analyzed leading to an estimate of the elapsed time per iteration. Based on this analysis, a heuristic algorithm is devised for distributing and splitting the blocks for a good static load balance. Speed-up results are presented for a wing, a complete aircraft and an air inlet.

Anisotropic grid adaptation for Navier--Stokes' equations

Navier-Stokes equations are discretized in space by a finite volume method. Error equations are derived which are approximately satisfied by the errors in the solution. The dependence of the solution errors on the discretization errors is analyzed in certain flow cases. The grid is adapted based on the estimated discretization errors. The refinement and coarsening of the grid are anisotropic in the sense that it is different in different directions in the computational domain. The adaptation algorithm is applied to laminar, viscous flow over a flat plate, in a channel with a bump, and around a cylinder and an airfoil.

On numerical errors in the boundary conditions of the Euler equations

Numerical errors in solution of the Euler equations of fluid flow are studied. The error equations are solved to analyze the propagation of the discretization errors. In particular, the errors caused by the boundary conditions and their propagation are investigated. Errors generated at a wall are transported differently in subsonic and supersonic flows. Accuracy may be lost due to the accumulation of errors along the walls. This can be avoided by increasing the accuracy of the boundary conditions. Large errors may still arise locally at the leading edge of a wing profile. There, a fine grid is the best way to reduce the error.

The GMRES method improved by securing fast wave propagation

Abstract The GMRES method applied directly to nonsymmetric systems cannot be expected to give good convergence propertities. The usual remedy is to introduce a preconditioning. For systems rising from first-order PDEs, no generally applicable preconditioner has been constructed. In this paper we take another approach by considering GMRES as an approximation of a time-dependent problem. In this way a modification can be constructed such that first wave propagation is secured. The resulting convergence improvement is demonstrated for certain problems arising from two-dimensional PDEs modelling compressible flow.