Lax [1971] has shown that for strictly hyperbolic systems of conservation laws for a vector function u{x ,t } all entropy inequalities of the form ( 11) are equivalent, provided that for each system of equations the entropy functions U(u ) that become considered strictly convex functions of u and the weak solutions u(x,t) of the conservation laws have only moderately strong discontinuities. Then we replace T in the resulting estimate and in (22) we get the estimate of the theorem. There are several other methods that can be considered modifications to the Godunov scheme.
Now consider the solution Uv of the difference equations (3} and interpolate it with the piecewise polynomial i;P(X) e c3 of deBoor's construction.This case can be solved by exponential scaling of some of the variables y(i) in the differential equation (2.0.1a) However, this tells us nothing about the limits of the solution of the system (32) - that information and thus the global estimate of the error ultimately depends on the condition of the system (32) and thus of the goodness of the original problem (8) over the entire interval [O,L].
More useful estimates 0C1>, oC2) in terms of the solution of the difference approximation (3) are given below. In this work, we consider a system of ordinary differential equations of the first order of the form. Now it remains to prove that the behavior of the breaking point of solution (1) is governed by the equations for w(r+l) and wCr+2) and that these equations.
In this appendix, we will briefly consider the case where the behavior of the solution (1). The behavior of solutions (1) can be understood by looking at the eigenvalues of the matrix. For example, Kreiss [1981] pointed out that the refinement of a grid that adapts to a solution may not solve the problem.
Numerical Methods for Problems in One Space Dimension
In Section 3.5 we apply the theory discussed in Chapter 2 to the solution of the equations for an isentropic thermal non-conducting gas in one spatial dimension. In this section we discuss difference approximations for a scalar second-order ordinary differential equation of the form However, the coefficients for the difference scheme are considered to be continuous functions of the coefficients of the differential equation.
In the first two numerical examples (figs. 1 and 2) the difference method of Section 3.2 was applied to a uniform mesh to demonstrate the monotonicity-preserving property of the method. A leftward shock forms on the left side of the solution and a rarefaction wave on the right. The speed of the moving coordinate system in such a case can be taken as such that in the resulting coordinate frame, a measure of the change of the solution with time is minimized.
At each of the subintervals thus defined, we recompute the solution using a combination of a finer grid and a moving coordinate system to reduce the error to an acceptable size. This coarse mesh is chosen so that smooth parts of the solution will be well resolved on it (while regions of rapid change will not necessarily be resolved). We chose to estimate the subinterval rate within this grid iteration process.
After each intermediate solution u(x,t+k) is computed from the solution at time t, u(x,t), we choose the velocity for the moving coordinate system by minimizing the discrete L2-norm of the change in the solution with respect to the velocity c, i.e. we are looking for an approximate solution c to. This grid is chosen in such a way that it is possible to resolve the smooth features of the solution well. Assume that the solution of the differential equation (for example 3.3.1) has been calculated exactly up to time t.
{If we need solution values at time t that are not on the grid, then they can be found. This allows the refinement subintervals to appear and disappear as the local smoothness of the solution changes. No attempt was made to approximate the dissipative term in the Lax-W endroff part of the solution.
The minimization process of estimating the local speed of the solution was done using a Golden Ratio technique. By linear combination of the equations (1), the third equation can be reduced to the form.
Numerical Methods for Problems in More Than One Space Dimension
A more rational approach to local mesh refinement is to embed locally oriented grids in the coarse grid and interpolate between the different grids when solving the differential equations. With a simple extension of B. Kreiss [1981's mesh generation approach], curvilinear grids could also be embedded in the coarse rectangular grid in such a way that a bump is resolved (see Figure 4.1.2c). The difference method we used for the one-dimensional case (see Section 2.3) is implicit in the time direction.
Let us assume that in the initial conditions Ua (x, y) there is a region of fast transition oriented obliquely to the grid. Because of the fast transition region, we would expect automatic refinement to occur so that the solution to each of these one-dimensional problems would be solved. We again expect automatic mesh refinement to occur in the region near the fast transition.
However, note that the right-hand side of the equation for u(xi,y,t) depends on the values of the solution u(xi,y) calculated in the previous step. It is this method that we will use for interpolation in two-dimensional problems. Reyna's interpolation procedure works well as long as the fast transition regions of the function f (x ,y) do not lie nearly parallel to the lines y = canst.
However, in the context of solving a differential equation in two space dimensions numerically, this case should not be a problem because it is the case of a shock oriented parallel to the mesh. In the remainder of this section, we will discuss and give numerical examples of the use of the method outlined above to resolve stationary fast transients directed obliquely to a mesh in solutions of the two-dimensional Burgers' equation (equation (1) ). The plus signs '+' indicate the locations of the mesh points in the final refined mesh.
Lines are also drawn in the direction perpendicular to the sweep direction (eg in the y direction in Figure 4.1.4a) to indicate the location of the underlying coarse mesh. The coarse mesh in this case was not uniform, but was finer near the center of the domain where the corner of the "wedge" occurs. Of course, due to the dissipative terms in Burgers' equation, the shock has a limited width, and the contact discontinuity becomes wider with time.
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