Rezone algorithms for the Burgers' equation



Principle ideas of our methodology

EMB rezone strategy
The rezone strategy aims to modify the current mesh (at time moment t = tn) such that the error at time moment t = tn+1 is minimized. This error is a superposition of an interpolation error, an error due to a time advancing method, and a space discretization error. If the grid is not modified, then the interpolation error is zero. On the other hand, the current grid may not be the best grid to represent new features of the solution after one time step. The goal of the EMB resize strategy is to achieve a balance (if possible) between the errors which minimizes the total error at time moment t = tn+1 over a class of smooth meshes.


Example I: viscous Burgers' equation
Let us consider the viscous Burgers equation
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subject to boundary and initial conditions such that an exact solution is known. Let T=0.9 and &epsilon=0.005. The trajectories of mesh nodes for the pure Lagrangian method, ALE method with the Reference Jacobian Matrix (RJM) rezone strategy, and ALE method with the EMB rezone strategy are shown in Figures a, b and c, respectively. Horizontal slices represent the mesh at the corresponding time moments. The red graphs are discrete solutions at times t=0 and t=T.
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In the pure Lagrangian method, the nodes get progressively closer to each other, that is, the size of each cell diminishes. In regions where solution has big gradient, the size of mesh cells diminishes faster. It confirms a well-known statement that the method of characteristics is not well-suited for general hyperbolic PDEs and in particular for Burgers' equation. From a practical viewpoint, constantly diminishing size means that a time step is goes to zero according to the stability condition and becomes so small that the calculation practically stalls. In the ALE method with the RJM rezone strategy, the mesh trajectories roughly follow the Lagrangian trajectories. The mesh is smooth and does not have problems of the pure Lagrangian mesh. However, main features of the solution are under-resolved. In the ALE method with the EMB rezone strategy, the mesh is smooth and resolves solution features at each time moment.

The left picture below demonstrates behavior of the L2-norm of error on meshes with 32 cells.
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The pure Lagrangian method is accurate up to about t=0.1. After this moment, dynamics of mesh cells do not correspond to dynamics of solution features and accuracy starts to degrade. For the RJM ALE method, the error first starts to grow because the method makes the mesh smoother (compared to the original mesh) and, by doing this, it reduces resolution of the features. Then, the error stays approximately constant. After t=0.3 the accuracy is significantly better then accuracy of the pure Lagrangian method. For the EMB ALE method, the error is about the same as for the pure Lagrangian method for t < 0.15, but after this moment, it is significantly less than the errors for the other two methods.

The right picture demonstrates the linear convergence of the error with respect to the number of mesh steps for both rezone strategies. However, for the given accuracy, e.g. 0.002, we need about 2 times less mesh points for the EMB ALE method then for the RJM ALE method. This difference grows when &epsilon decreases and the solution profile becomes more sharp.


Movies of 2D moving meshes

The first 2D experiments with the EMB rezone strategy have been done for analytic functions which resemble solutions of a viscous Burgers' equation.

1. Two parallel viscous waves (mpeg file, 4.3Mb)
2. Two orthogonal viscous waves (mpeg file, 3.5Mb)


References.

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