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A robust first order meshfree method for time-dependent nonlinear conservation laws

Samuel Kwan, Jesse Chan

Abstract

We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts differentiations. We describe how to efficiently construct such operators on a point cloud. We then study the performance of such differentiations, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws, with focus on the advection equation and the compressible Euler equations. We observe numerically that, while the resulting mesh-free differentiation operators are only $O(h^\frac{1}{2})$ accurate in the $L^2$ norm, they achieve $O(h)$ rates of convergence when applied to the numerical solution of PDEs.

A robust first order meshfree method for time-dependent nonlinear conservation laws

Abstract

We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts differentiations. We describe how to efficiently construct such operators on a point cloud. We then study the performance of such differentiations, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws, with focus on the advection equation and the compressible Euler equations. We observe numerically that, while the resulting mesh-free differentiation operators are only accurate in the norm, they achieve rates of convergence when applied to the numerical solution of PDEs.

Paper Structure

This paper contains 19 sections, 1 theorem, 52 equations, 11 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Summation by parts operators
  3. Constructing meshfree SBP Operators
  4. Boundary operators
  5. Algebraic construction of the volume SBP operator
  6. Optimization of the norm matrix
  7. Accuracy of the optimized norm matrix
  8. On the observed convergence rates
  9. Finding Suitable Notion of Adjacency
  10. Details on the computational implementation
  11. A robust first order accurate meshfree method
  12. Notation
  13. Discretization
  14. Stabilization
  15. Numerical results
  16. ...and 4 more sections

Key Result

Lemma 1

For $\bm{\mathsf{ b}} = -\frac{1}{2}\bm{\mathsf{ E}}_x\bm{\mathsf{ 1}}$, where $\bm{\mathsf{ E}}_x$ satisfies prop3b, $\bm{\mathsf{ L\bm{\mathsf{ \Psi}}}} = \bm{\mathsf{ b}}$ has a solution.

Figures (11)

  • Figure 1: The domain $\Omega_1$ for $n_x = n_y = 25, n_b = 75, R = 3$
  • Figure 2: Absolute values of the computed errors for the differential operators on $u_1$ and $u_2$ on Grid 3 using the "Euclidean Radius" adjacency method.
  • Figure 3: The domain $\Omega_2$ for $n_x = n_y = 25, n_b = 75, n_i = 30, R = 3$. Here, $n_i$ denotes the number of nodes on the boundary of each inner circle.
  • Figure 4: Absolute value errors for differential operators on $u_1$ and $u_2$ on $\Omega_2$ (Euclidean Radius, Grid 3).
  • Figure 5: The computed pressure using the HLLC flux at various times. The color limits are taken from 0.008 to 0.25.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Remark 1
  • Remark 2