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Efficiency of a Moving Mesh System with a Curvature-type Monitor Applied to Burgers’ Equation

Marianne DeBrito, Annaliese Keiser, Taima Younes Mentor: Joan Remski January 26, 2019

This research was conducted at the University of Michigan-Dearborn, and this project was supported by the National Science Foundation (DMS-1659203), the National Security Agency, and the University of Michigan-Dearborn. DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 1 / 18

Outline

• 1. Burgers’ Equation
• 2. Physical Solution PDE & Errors
• 3. Moving Mesh PDE & Benefits
• 4. Our Theorem
• 5. Why it matters

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 2 / 18

An Interesting RDM: Burgers’ Equation

Simplified Navier-Stokes equation, in 1-D: ut = ǫuxx − (1 2u2)x

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 3 / 18

An Interesting RDM: Burgers’ Equation

Simplified Navier-Stokes equation, in 1-D: ut = ǫuxx − (1 2u2)x Initial conditions: u(x,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 x ≤ 0.25 2 − 4x 0.25 < x ≤ 0.5 x > 0.5

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 3 / 18

An Interesting RDM: Burgers’ Equation

Simplified Navier-Stokes equation, in 1-D: ut = ǫuxx − (1 2u2)x Initial conditions: u(x,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 x ≤ 0.25 2 − 4x 0.25 < x ≤ 0.5 x > 0.5 Boundary conditions: u(0,t) = 1, u(1,t) = 0

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 3 / 18

An Interesting RDM: Burgers’ Equation

Simplified Navier-Stokes equation, in 1-D: ut = ǫuxx − (1 2u2)x Initial conditions: u(x,0) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 x ≤ 0.25 2 − 4x 0.25 < x ≤ 0.5 x > 0.5 Boundary conditions: u(0,t) = 1, u(1,t) = 0 Propagating wavefront with steepness controlled by ǫ

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 3 / 18

Evolution of a Numerical Solution to Burgers’ Equation Over Time (ǫ = 0.01)

ut = ǫuxx − (1 2u2)x

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 4 / 18

Evolution of a Numerical Solution to Burgers’ Equation Over Time (ǫ = 0.001)

ut = ǫuxx − (1 2u2)x

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 5 / 18

Approximating Solutions over Time

Finding u(xj,tn+1):

uj,n+1 = ⎛ ⎜ ⎝ ǫ∆t h2

j

⎞ ⎟ ⎠ uj−1,n + ⎛ ⎜ ⎝ 1 − 2 ǫ∆t h2

j

⎞ ⎟ ⎠ uj,n + ⎛ ⎜ ⎝ ǫ∆t h2

j

⎞ ⎟ ⎠ uj+1,n + ∆t 4hj (u2

j+1,n − u2 j−1,n) + uj,n

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 6 / 18

Moving Mesh Methods

Introduction to Moving Mesh Methods

Adaptive techniques to solve partial differential equations numerically As physical solution, u, evolves, so do the grid points, xj

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 7 / 18

Moving Mesh Methods

Introduction to Moving Mesh Methods

Adaptive techniques to solve partial differential equations numerically As physical solution, u, evolves, so do the grid points, xj Goal Balance the undesirable characteristics of the physical PDE by adjusting points using a moving mesh PDE.

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 7 / 18

Moving Mesh Methods

The Moving Mesh Equation

Moving Mesh PDE: xt = (ωxξ)ξ for x = x(ξ,t) Steady State Moving Mesh PDE: 0 = (ωxξ)ξ ω = Monitor Function, aka the “Mesh Density Function”

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 8 / 18

Moving Mesh Methods

Mesh Movement Mapping

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 9 / 18

Moving Mesh Methods

Examples of Moving Mesh

Figure: A fixed mesh method compared to an Arc Length-type mesh

ω = √ (1 + αu2

x)

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 10 / 18

Moving Mesh Methods

Examples of Moving Mesh

Figure: A fixed mesh method compared to a Curvature-type mesh

ω = (1 + ǫpu2

xx)1/q

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 11 / 18

Analytical Results

Effectiveness of the Curvature Monitor

Here, note that for z to be O(C) means that M1C ≤ z ≤ M2C, where M1 and M2 are arbitrary constants. Theorem (DKRY’18) Let u = u(x) be the physical solution that satisfies the following assumptions: (i) the solution has large gradient in Ωǫ, i.e., ∣∣ux∣∣∞ = O(ǫ−1) and in [0,1] ∖ Ωǫ ∣∣ux∣∣∞ = O(1), and (ii) the solution has large curvature over Ωǫ, i.e., ∣∣uxx∣∣∞ = O(ǫ−2) and in [0,1] ∖ Ωǫ ∣∣uxx∣∣∞ = O(1), where meas(Ωǫ) = O(ǫ).

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 12 / 18

Analytical Results

Effectiveness of the Curvature Monitor

Here, note that for z to be O(C) means that M1C ≤ z ≤ M2C, where M1 and M2 are arbitrary constants. Theorem (DKRY’18) Let u = u(x) be the physical solution that satisfies the following assumptions: (i) the solution has large gradient in Ωǫ, i.e., ∣∣ux∣∣∞ = O(ǫ−1) and in [0,1] ∖ Ωǫ ∣∣ux∣∣∞ = O(1), and (ii) the solution has large curvature over Ωǫ, i.e., ∣∣uxx∣∣∞ = O(ǫ−2) and in [0,1] ∖ Ωǫ ∣∣uxx∣∣∞ = O(1), where meas(Ωǫ) = O(ǫ). Then, with the monitor function ω = (1 + ǫpu2

xx)1/q,

where ǫ ≤ 1, p and q are nonnegative and p + q ≥ 4, the solution in computational domain, v(ξ) = u(x(ξ)), and the mapping from the physical domain to the computational domain, ξ = ξ(x), satisfy the following bounds:

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 12 / 18

Analytical Results

Effectiveness of the Curvature Monitor

Here, note that for z to be O(C) means that M1C ≤ z ≤ M2C, where M1 and M2 are arbitrary constants. Theorem (DKRY’18) Let u = u(x) be the physical solution that satisfies the following assumptions: (i) the solution has large gradient in Ωǫ, i.e., ∣∣ux∣∣∞ = O(ǫ−1) and in [0,1] ∖ Ωǫ ∣∣ux∣∣∞ = O(1), and (ii) the solution has large curvature over Ωǫ, i.e., ∣∣uxx∣∣∞ = O(ǫ−2) and in [0,1] ∖ Ωǫ ∣∣uxx∣∣∞ = O(1), where meas(Ωǫ) = O(ǫ). Then, with the monitor function ω = (1 + ǫpu2

xx)1/q,

where ǫ ≤ 1, p and q are nonnegative and p + q ≥ 4, the solution in computational domain, v(ξ) = u(x(ξ)), and the mapping from the physical domain to the computational domain, ξ = ξ(x), satisfy the following bounds: ∣∣xξ∣∣∞ = O(1), ∣∣ξx∣∣∞ = O(ǫ

p−4 q ),

and 0 ≤ ∣∣vξ∣∣∞ ≤ Mǫ

4−p−q q

.

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 12 / 18

Analytical Results

Corollary When Considering a Discrete System

Corollary (DKRY’18) When considering the system discretely, with the same hypotheses as previously, where hj = xj+1 − xj, the following bounds are satisfied: (i) On [0,1] ∖ Ωǫ minhj = O(∆ξ) (ii) On Ωǫ: minhj = O(ǫ

4−p q ∆ξ) DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 13 / 18

Analytical Results

Mistakes Were Made: Types of Errors

Truncation error: ux(xj) = u(xj+1) − u(xj−1) 2hj + (2hj)2uxx(xj) + ... When uxx is large (we assume O(ǫ−2)), we need hj very small A fixed mesh uses hj = ∆ξ

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 14 / 18

Analytical Results

Mistakes Were Made: Types of Errors

Truncation error: ux(xj) = u(xj+1) − u(xj−1) 2hj + (2hj)2uxx(xj) + ... When uxx is large (we assume O(ǫ−2)), we need hj very small A fixed mesh uses hj = ∆ξ A moving mesh uses minhj = O(∆ξ ǫ

4−p q ) . . . p = 1,q = 6 DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 14 / 18

Analytical Results

Mistakes Were Made: Types of Errors

Truncation error: ux(xj) = u(xj+1) − u(xj−1) 2hj + (2hj)2uxx(xj) + ... When uxx is large (we assume O(ǫ−2)), we need hj very small A fixed mesh uses hj = ∆ξ A moving mesh uses minhj = O(∆ξ ǫ

4−p q ) . . . p = 1,q = 6

On a fixed mesh, the truncation error is of order ∆ξ2ǫ−2, but on this moving mesh system, truncation error is of order ∆ξ2ǫ−1

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 14 / 18

Example Application and Evidence

Example of Moving Mesh on Burgers’ Equation

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 15 / 18

Example Application and Evidence

Mesh Trajectories for the Modeled Solution

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 16 / 18

Example Application and Evidence

Numerical Evidence for an Approximated Solution of Burgers’ Equation

Table: ǫ values for ω = (1 + ǫu2

xx)1/6

ǫ ∣∣ux∣∣∞ ∣∣vξ∣∣∞ exp minhj exp ∣∣ξx∣∣∞ exp 0.01 12.449 3.193

• 0.448

0.00319 0.713 5.220

• 0.713

0.005 24.919 4.357

• 0.451

0.00195 0.665 8.556

• 0.658

0.0025 49.495 5.958

• 0.452

0.00123 0.611 13.498

• 0.615

0.00125 97.907 8.152

• 0.451

0.00080 0.568 20.672

• 0.567

0.000625 193.055 11.145 0.00054 30.606 0 ≤ ∣∣vξ∣∣∞ ≤ Mǫ

−1 2 , minhj = O(ǫ 1 2 )

and ∣∣ξx∣∣∞ = O(ǫ

−1 2 ).

DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 17 / 18

Example Application and Evidence

Any Questions?

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Thank You!

Special Thanks: To Joan Remski, for being our wonderful mentor; to the University of Michigan-Dearborn REU Site in Mathematical Analysis, Algebraic Music Theory, and their Applications for hosting our research; and to the National Science Foundation and the National Security Agency for funding

• ur REU.

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