Burton Wendroff

Two-phase flow: Models and methods

Abstract A variety of two-phase flow models can be derived following a few basic principles, which are here illustrated which no more generality than is essential. Among the models derived is one already widely used in applications, even though it is ill-posed in the sense of Hadamard. Final assessment of such models remains a distant goal, but will clearly involve numerical solutions; several methods in current use are discussed with a guide to selecting the one appropriate to a particular problem.

Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations

The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

Composite Schemes for Conservation Laws

Global composition of several time steps of the two-step Lax--Wendroff scheme followed by a Lax--Friedrichs step seems to enhance the best features of both, although it is only first order accurate. We show this by means of some examples of one-dimensional shallow water flow over an obstacle. In two dimensions we present a new version of Lax--Friedrichs and an associated second order predictor-corrector method. Composition of these schemes is shown to be effective and efficient for some two-dimensional Riemann problems and for Nohs infinite strength cylindrical shock problem. We also show comparable results for composition of the predictor-corrector scheme with a modified second order accurate weighted essentially nonoscillatory (WENO) scheme. That composition is second order but is more efficient and has better symmetry properties than WENO alone. For scalar advection in two dimensions the optimal stability of the new predictor-corrector scheme is shown using computer algebra. We also show that the generalization of this scheme to three dimensions is unstable, but by using sampling we are able to show that the composites are suboptimally stable.

Supra-convergent schemes on irregular grids

As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencils points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a ( k/2J -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerovs tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order con- vergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three- periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.

The Relative Efficiency of Finite Difference and Finite Element Methods. I: Hyperbolic Problems and Splines

The finite element method, using smooth splines as basis functions, applied to the model problem

Convergence Estimates for Galerkin Methods for Variable Coefficient Initial Value Problems

u_t = cu_x

An efficient linearity-and-bound-preserving remapping method

with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.

Potential Vorticity Evolution of a Protoplanetary Disk with an Embedded Protoplanet

The use of Galerkin’s method for the approximate solution of the initial value problem for certain simple equations

Two-dimensional shallow water equations by composite schemes

{{\partial u} / {\partial t = Pu}}

Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian methods

, where P is a differential operator of order m with respect to x, is analyzed when the approximate solution is sought in the space of smooth splines of order