Schemes with Well-Controlled Dissipation (WCD) Jan Ernest Seminar of Applied Mathematics (SAM), ETH Z¨ urich, Switzerland Jan Ernest WCD Schemes
Joint work with ◮ Siddhartha Mishra ◮ Center of Mathematics for Applications (CMA), University of Oslo, Norway. ◮ Philippe G. LeFloch ◮ Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Universit´ e Pierre et Marie Curie, Paris, France. Jan Ernest WCD Schemes
Scope of the Talk Hyperbolic conservation law u t + f ( u ) x = 0 , u = u ( x , t ) , ( x , t ) ∈ R × R + . Jan Ernest WCD Schemes
Scope of the Talk Hyperbolic conservation law with small-scale effects u ε = u ε ( x , t ) , u ε t + f ( u ε ) x = R ε ( u ε ) , ( x , t ) ∈ R × R + . Interested in limit as ε → 0. Jan Ernest WCD Schemes
Scope of the Talk Two cases: 1. diffusive-dispersive regularization R ε ( u ) = ε u xx + δε 2 u xxx Jan Ernest WCD Schemes
Scope of the Talk Two cases: 1. diffusive-dispersive regularization R ε ( u ) = ε u xx + δε 2 u xxx 2. pseudo-parabolic regularization R ε ( u ) = ε u xx + δε 2 u xxt Jan Ernest WCD Schemes
Scope of the Talk Two cases: 1. diffusive-dispersive regularization R ε ( u ) = ε u xx + δε 2 u xxx 2. pseudo-parabolic regularization R ε ( u ) = ε u xx + δε 2 u xxt Goal: Approximate solutions for fixed δ and ε → 0 numerically. Jan Ernest WCD Schemes
Classical vs. Nonclassical Solutions Why interesting? ◮ Classical and Nonclassical shock waves may arise depending on value of δ Jan Ernest WCD Schemes
Classical vs. Nonclassical Solutions Why interesting? ◮ Classical and Nonclassical shock waves may arise depending on value of δ ◮ Classical solutions ◮ satisfy Lax / Oleinik entropy inequalities ◮ S ( u ) t + Q ( u ) x ≤ 0 for all entropy pairs ( S , Q ) ◮ are TVD Jan Ernest WCD Schemes
Classical vs. Nonclassical Solutions Why interesting? ◮ Classical and Nonclassical shock waves may arise depending on value of δ ◮ Classical solutions ◮ satisfy Lax / Oleinik entropy inequalities ◮ S ( u ) t + Q ( u ) x ≤ 0 for all entropy pairs ( S , Q ) ◮ are TVD ◮ Nonclassical solutions ◮ do not satisfy Lax / Oleinik entropy conditions ◮ satisfy S ( u ) t + Q ( u ) x ≤ 0 for a single entropy pair ( S , Q ) ◮ not TVD ◮ additional criterion for uniqueness: Kinetic relation (Hayes & LeFloch (1997)) ◮ Controls entropy dissipation at a shock as a function of the shock speed. Jan Ernest WCD Schemes
Example: Cubic Conservation Law Regularized cubic conservation law: � u 3 � x = ε u xx + δε 2 u xxx u t + Jacobs, McKinney, Shearer (1995): ◮ Systematic study of traveling wave solutions ◮ Analytic expressions for nonclassical solutions Jan Ernest WCD Schemes
Cubic CL: Classical Solution Classical solution ( δ = 0): � u 3 � u t + x = ε u xx Jan Ernest WCD Schemes
Cubic CL: Classical Solution Classical solution ( δ = 0): � u 3 � u t + x = ε u xx 5 0 −5 −0.5 0 0.5 1 1.5 2 2.5 (a) t = 0 Jan Ernest WCD Schemes
Cubic CL: Classical Solution Classical solution ( δ = 0): � u 3 � u t + x = ε u xx 5 5 0 0 −5 −5 −0.5 0 0.5 1 1.5 2 2.5 −0.5 0 0.5 1 1.5 2 2.5 (a) t = 0 (b) t = 0 . 05 Jan Ernest WCD Schemes
Cubic CL: Nonclassical Solution Nonclassical solution ( δ = 1): � u 3 � x = ε u xx + ε 2 u xxx u t + Jan Ernest WCD Schemes
Cubic CL: Nonclassical Solution Nonclassical solution ( δ = 1): � u 3 � x = ε u xx + ε 2 u xxx u t + 5 0 −5 −0.5 0 0.5 1 1.5 2 2.5 (a) t = 0 Jan Ernest WCD Schemes
Cubic CL: Nonclassical Solution Nonclassical solution ( δ = 1): � u 3 � x = ε u xx + ε 2 u xxx u t + 5 5 0 0 −5 −5 −0.5 0 0.5 1 1.5 2 2.5 −0.5 0 0.5 1 1.5 2 2.5 (a) t = 0 (b) t = 0 . 05 Jan Ernest WCD Schemes
Numerical Approximation Standard schemes fail to approximate nonclassical solutions completely. They always converge to the classical solution ( δ = 0). 5 Lax−Friedrichs Rusanov Exact 0 −5 0 0.5 1 1.5 2 ⇒ Design schemes that approximate correct solution for δ � = 0. Jan Ernest WCD Schemes
Approaches to Resolve Nonclassical Shocks Numerical solvers for regularized cubic conservation law: ◮ Kissling & Rohde (2010): Heterogeneous multiscale approach ◮ LeFloch & Mohammadian (2008): High-order finite difference schemes with controlled dissipation � j = p � j = p � j = p � � � + δε 2 du i dt = − 1 ε � α j ( u 3 ) i + j � � + β j u i + j γ j u i + j (∆ x ) 2 (∆ x ) 3 ∆ x j = − p j = − p j = − p ◮ α j , β j and γ j chosen such that scheme is of order 2 p and leading order terms in equivalent equation match the small-scale effects. Jan Ernest WCD Schemes
Failure at Strong Shocks Observation: Approximations for ε = 5 · 10 − 3 dx . 4 2 0 −2 −4 0.8 1 1.2 1.4 (a) u L = 4 Jan Ernest WCD Schemes
Failure at Strong Shocks Observation: Approximations for ε = 5 · 10 − 3 dx . 20 4 15 10 2 5 0 0 −5 −2 −10 −15 −4 0.8 1 1.2 1.4 0.95 1 1.05 1.1 (a) u L = 4 (b) u L = 14 Jan Ernest WCD Schemes
Failure at Strong Shocks Observation: Approximations for ε = 5 · 10 − 3 dx . 20 4 20 15 15 10 2 10 5 5 0 0 0 −5 −5 −2 −10 −10 −15 −15 −4 0.8 1 1.2 1.4 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 (a) u L = 4 (b) u L = 14 (c) u L = 18 Jan Ernest WCD Schemes
Failure at Strong Shocks Observation: Approximations for ε = 5 · 10 − 3 dx . 20 4 20 15 15 10 2 10 5 5 0 0 0 −5 −5 −2 −10 −10 −15 −15 −4 0.8 1 1.2 1.4 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 (a) u L = 4 (b) u L = 14 (c) u L = 18 ◮ Scheme performs well for small shocks but fails for large shocks. Jan Ernest WCD Schemes
Construction of Schemes with Well-Controlled Dissipation Cubic conservation law: � u 3 � x = ε u xx + δε 2 u xxx u t + Idea 1 (Controlled Dissipation): Develop schemes such that leading-order terms of equivalent equation match the regularizing terms exactly . j = p j = p j = p + δ c 2 du i dt = − 1 + c � � � α j ( u 3 ) i + j β j u i + j γ j u i + j ∆ x ∆ x ∆ x j = − p j = − p j = − p Jan Ernest WCD Schemes
Construction of Schemes with Well-Controlled Dissipation Cubic conservation law: � u 3 � x = ε u xx + δε 2 u xxx u t + Idea 1 (Controlled Dissipation): Develop schemes such that leading-order terms of equivalent equation match the regularizing terms exactly . j = p j = p j = p + δ c 2 du i dt = − 1 + c � � � α j ( u 3 ) i + j β j u i + j γ j u i + j ∆ x ∆ x ∆ x j = − p j = − p j = − p Jan Ernest WCD Schemes
Equivalent Equation Cubic conservation law: � u 3 � x = ε u xx + δε 2 u xxx u t + Equivalent equation: u t = − ( u 3 ) x + c ∆ xu xx + δ c 2 ∆ x 2 u xxx + higher order terms � �� � � �� � (h.o.t.) leading order terms (l.o.t.) Jan Ernest WCD Schemes
Equivalent Equation Cubic conservation law: � u 3 � x = ε u xx + δε 2 u xxx u t + Equivalent equation: δε 2 u xxx ε u xx � �� � � �� � u t = − ( u 3 ) x + δ c 2 ∆ x 2 u xxx c ∆ xu xx + + higher order terms � �� � � �� � (h.o.t.) leading order terms (l.o.t.) Jan Ernest WCD Schemes
Recall: Failure at Strong Shocks 20 4 20 15 15 10 2 10 5 5 0 0 0 −5 −5 −2 −10 −10 −15 −15 −4 0.8 1 1.2 1.4 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 (a) u L = 4 (b) u L = 14 (c) u L = 18 ◮ Scheme performs well for small shocks but fails for large shocks. Jan Ernest WCD Schemes
Equivalent Equation Equivalent equation: u t = − ( u 3 ) x + c ∆ xu xx + δ c 2 ∆ x 2 u xxx + higher order terms � �� � � �� � (h.o.t.) leading order terms (l.o.t.) ◮ For large shocks the higher-order terms (h.o.t.) in the equivalent equation start to influence the approximation. Jan Ernest WCD Schemes
Schemes with Well-Controlled Dissipation Idea 2: Impose a balance between leading-order and higher-order terms of the equivalent equation. Want: | h . o . t . | < τ | l . o . t . | for τ << 1 . ◮ Ensures that the nonclassical behaviour in approximation mostly comes from the correct small-scale mechanisms Jan Ernest WCD Schemes
Derivation of the WCD Condition Scheme: � j = p � j = p � j = p � � � + δ c 2 du i dt = − 1 c � � � α j f i + j + β j u i + j γ j u i + j ∆ x ∆ x ∆ x j = − p j = − p j = − p Equivalent equation: u t = − f ( u ) x + c ∆ xu xx + δ c 2 ∆ x 2 u xxx ∞ ∆ x k − 1 ∞ ∆ x k − 1 ∞ ∆ x k − 1 k f [ k ] + c k u [ k ] + δ c 2 � A p � B p � C p k u [ k ] − k ! k ! k ! k =2 p +1 k =2 p +1 k =2 p +1 with p p p A p � α j j k , B p � β j j k , C p � γ j j k k = k = k = j = − p j = − p j = − p Jan Ernest WCD Schemes
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