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: u t = ǫ u xx − ( 1 2 u 2 ) 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: u t = ǫ u xx − ( 1 2 u 2 ) x Initial conditions: ⎧ 1 x ≤ 0 . 25 ⎪ ⎪ ⎪ ⎪ u ( x , 0 ) = ⎨ 2 − 4 x 0 . 25 < x ≤ 0 . 5 ⎪ ⎪ ⎪ ⎪ x > 0 . 5 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: u t = ǫ u xx − ( 1 2 u 2 ) x Initial conditions: ⎧ 1 x ≤ 0 . 25 ⎪ ⎪ ⎪ ⎪ u ( x , 0 ) = ⎨ 2 − 4 x 0 . 25 < x ≤ 0 . 5 ⎪ ⎪ ⎪ ⎪ x > 0 . 5 0 ⎩ 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: u t = ǫ u xx − ( 1 2 u 2 ) x Initial conditions: ⎧ 1 x ≤ 0 . 25 ⎪ ⎪ ⎪ ⎪ u ( x , 0 ) = ⎨ 2 − 4 x 0 . 25 < x ≤ 0 . 5 ⎪ ⎪ ⎪ ⎪ x > 0 . 5 0 ⎩ 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) u t = ǫ u xx − ( 1 2 u 2 ) 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) u t = ǫ u xx − ( 1 2 u 2 ) x DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 5 / 18
Approximating Solutions over Time Finding u ( x j , t n + 1 ) : ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ǫ ∆ t 1 − 2 ǫ ∆ t ǫ ∆ t u j + 1 , n + ∆ t ( u 2 j + 1 , n − u 2 u j , n + 1 = ⎜ ⎟ u j − 1 , n + ⎜ ⎟ u j , n + ⎜ ⎟ j − 1 , n ) + u j , n h 2 h 2 h 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 4 h j j j j 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, x j 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, x j 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: x t = ( ω x ξ ) ξ for x = x ( ξ, t ) Steady State Moving Mesh PDE: 0 = ( ω x ξ ) 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 √ ( 1 + α u 2 ω = 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 + ǫ p u 2 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 M 1 C ≤ z ≤ M 2 C , where M 1 and M 2 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 Ω ǫ , i.e., ∣∣ u x ∣∣ ∞ = O ( ǫ − 1 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u x ∣∣ ∞ = O ( 1 ) , and (ii) the solution has large curvature over Ω ǫ , i.e., ∣∣ u xx ∣∣ ∞ = O ( ǫ − 2 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u xx ∣∣ ∞ = O ( 1 ) , where meas ( Ω ǫ ) = 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 M 1 C ≤ z ≤ M 2 C , where M 1 and M 2 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 Ω ǫ , i.e., ∣∣ u x ∣∣ ∞ = O ( ǫ − 1 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u x ∣∣ ∞ = O ( 1 ) , and (ii) the solution has large curvature over Ω ǫ , i.e., ∣∣ u xx ∣∣ ∞ = O ( ǫ − 2 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u xx ∣∣ ∞ = O ( 1 ) , where meas ( Ω ǫ ) = O ( ǫ ) . Then, with the monitor function ω = ( 1 + ǫ p u 2 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 M 1 C ≤ z ≤ M 2 C , where M 1 and M 2 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 Ω ǫ , i.e., ∣∣ u x ∣∣ ∞ = O ( ǫ − 1 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u x ∣∣ ∞ = O ( 1 ) , and (ii) the solution has large curvature over Ω ǫ , i.e., ∣∣ u xx ∣∣ ∞ = O ( ǫ − 2 ) and in [ 0 , 1 ] ∖ Ω ǫ ∣∣ u xx ∣∣ ∞ = O ( 1 ) , where meas ( Ω ǫ ) = O ( ǫ ) . Then, with the monitor function ω = ( 1 + ǫ p u 2 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: p − 4 q ) , ∣∣ x ξ ∣∣ ∞ = O ( 1 ) , ∣∣ ξ x ∣∣ ∞ = O ( ǫ 4 − p − q and 0 ≤ ∣∣ v ξ ∣∣ ∞ ≤ M ǫ . 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 h j = x j + 1 − x j , the following bounds are satisfied: (i) On [ 0 , 1 ] ∖ Ω ǫ min h j = O ( ∆ ξ ) 4 − p q ∆ ξ ) (ii) On Ω ǫ : min h j = O ( ǫ DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 13 / 18
Analytical Results Mistakes Were Made: Types of Errors � Truncation error: u x ( x j ) = u ( x j + 1 ) − u ( x j − 1 ) + ( 2 h j ) 2 u xx ( x j ) + ... 2 h j When u xx is large (we assume O ( ǫ − 2 ) ), we need h j very small A fixed mesh uses h j = ∆ 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: u x ( x j ) = u ( x j + 1 ) − u ( x j − 1 ) + ( 2 h j ) 2 u xx ( x j ) + ... 2 h j When u xx is large (we assume O ( ǫ − 2 ) ), we need h j very small A fixed mesh uses h j = ∆ ξ 4 − p q ) . . . p = 1 , q = 6 A moving mesh uses min h j = O ( ∆ ξ ǫ DeBrito, Keiser, Remski, Younes Efficiency of Moving Mesh Methods January 26, 2019 14 / 18
Analytical Results Mistakes Were Made: Types of Errors � Truncation error: u x ( x j ) = u ( x j + 1 ) − u ( x j − 1 ) + ( 2 h j ) 2 u xx ( x j ) + ... 2 h j When u xx is large (we assume O ( ǫ − 2 ) ), we need h j very small A fixed mesh uses h j = ∆ ξ 4 − p q ) . . . p = 1 , q = 6 A moving mesh uses min h j = O ( ∆ ξ ǫ On a fixed mesh, the truncation error is of order ∆ ξ 2 ǫ − 2 , but on this moving mesh system, truncation error is of order ∆ ξ 2 ǫ − 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
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