Maps & Tents. Jay Gopalakrishnan. Portland State University. in collaboration with P. Monk, J. Schöberl, P. Sepúlveda, C.
|
|
- Homer Harrison
- 5 years ago
- Views:
Transcription
1 Maps & Tents Jay Gopalakrishnan Portland State University in collaboration with P. Monk, J. Schöberl, P. Sepúlveda, C. Wintersteiger IMA, Minneapolis, June 2017 Jay Gopalakrishnan 1/46
2 Who are we celebrating? I am the one on the left! [Source: www-users.math.umn.edu/ bcockbur/] Jay Gopalakrishnan 2/46
3 His 4 golden rules FUN NEW DOABLE IMPORTANT Jay Gopalakrishnan 3/46
4 Outline Introduction to tent pitching Basic idea History & background A one-dimensional example One-dimensional wave equation New mathematics for tent spaces General multidimensional hyperbolic systems Multidimensional tents Advantages & drawbacks Introducing MTP schemes Maps Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 4/46
5 Standard time stepping 1 Discretize all spatial differential operators. 2 Time step explicitly or implicitly using a time discretization. t Jay Gopalakrishnan 5/46
6 Standard time stepping 1 Discretize all spatial differential operators. 2 Time step explicitly or implicitly using a time discretization. t Goal of this talk: Explore an alternate time advance mechanism for hyperbolic problems. Jay Gopalakrishnan 5/46
7 Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t light cone initial data given x Jay Gopalakrishnan 6/46
8 Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t tent pole initial data given Pitch a tent x Jay Gopalakrishnan 6/46
9 Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t Pitch a tent initial data given x Problem is solvable in the tent provided the tent pole is not too high. Jay Gopalakrishnan 6/46
10 Spacetime tents Suppose we want to solve a hyperbolic problem in one space dimension, with finite propagation speed. t initial data given x Problem is solvable in the tent provided the tent pole is not too high. Jay Gopalakrishnan 6/46
11 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t x Jay Gopalakrishnan 7/46
12 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t t = 1 c (c = maximal characteristic speed) x Jay Gopalakrishnan 7/46
13 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ t = 1 c (c = maximal characteristic speed) Local CFL-condition: τ < 1 c x Jay Gopalakrishnan 7/46
14 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
15 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
16 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
17 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
18 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
19 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
20 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
21 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
22 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
23 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
24 Tent pitching in spacetime Time marching is possible, even with unstructured spatial mesh and varying time steps at different locations. t Advancing front τ x Jay Gopalakrishnan 7/46
25 Ancestry of Tent Pitching Ideas to advance in time by local operations in spacetime regions: [Richter 1994] Explicit elements using DG for wave eq. [Lowrie+Roe+van Leer 1995] Spacetime discontinuous Galerkin (SDG) method [Falk+Richter 1999] Explicit DG elements for some Friedrichs systems [Yin+Acharia+Sobh+Haber+Tortorelli 2000] SDG method for elastodynamics [Üngör+Sheffer 2000] Meshing by pitching tents in spacetime [Palaniappan+Haber+Jerrard 2004] SDG for conservation laws [Erickson+Guoy+Sullivan+Üngör 2005] Improved Tent Pitcher algorithm [Monk+Richter 2005] Wave equation in 2D using tent-pitched meshes and SDG Jay Gopalakrishnan 8/46
26 Next Introduction to tent pitching ! Basic idea ! History & background ! A one-dimensional example One-dimensional wave equation New mathematics for tent spaces General multidimensional hyperbolic systems Multidimensional tents Advantages & drawbacks Introducing MTP schemes Maps Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 9/46
27 Wave operator Wave equation: Let K R 2 be a spacetime domain. c 2 tt φ xx φ = f, on K. Rewrite: Set u 1 = c x φ and u 2 = t φ. [ ] [ ] [ ] [ ] t u 1 0 c x u 1 0 = t u 2 c 0 x u 2 c 2. f }{{}}{{} Au F Wave operator: A [ u1 u 2 A is a first order hyperbolic operator. ] [ ] t u = 1 c x u 2. t u 2 c x u 1 Jay Gopalakrishnan 10/46
28 The wave problem on a tent We want to solve the wave equation using tent pitching. The spacetime tent K can be of various shapes: (interior) K (left boundary) K K (right boundary) On each tent, we must discretize and solve Au = f the wave equation on a tent K, u g V with boundary & initial conditions (b.i.c.). Here V is the correct space with homogeneous b.i.c. Jay Gopalakrishnan 11/46
29 Space and trace Operator: A [ u1 u 2 ] [ ] t u = 1 c x u 2. t u 2 c x u 1 Graph space: Modern Friedrichs theory applies for A. [Jensen 2004], [Ern+Guermond+Caplain 2007] W = {v L 2 (K) 2 : Av L 2 (K) 2 }. Problem formulation: Au = f on a spacetime tent K, u g V b.i.c. on K. To say what V is, we need trace theory for W. Jay Gopalakrishnan 12/46
30 Diagonalize & simplify To understand the wave operator [ ] [ ] [ ] u1 0 c u1 Au = t u 2 c 0 x u 2 [ ] [ ] u1 = t QΛQ t u1 u x, 2 u 2 Here Λ = [ ] c 0, 0 c Q = 1 2 [ it suffices to understand the diagonalized (decoupled) operator D = Q t AQ: [ ] [ ] [ ] w1 D = Q t w1 tw 1 + c xw 1 AQ =. w 2 w 2 tw 2 c xw 2 Simplification! It suffices to study the graph space of the advection operator acting on scalar functions w. Âw = t w + α x w ]. Jay Gopalakrishnan 13/46
31 Advection operator The advection operator  = t + α x is also a Friedrichs operator. Let its graph space be Ŵ. Traces of Ŵ were studied earlier. [Bardos 1970], [Ern+Guermond 2004] Known trace result for advection If w is in Ŵ and dist(γ i, Γ o ) > 0, then the inflow and outflow traces w Γi, w Γo are in L 2 (Γ i ) and L 2 (Γ o ), respectively. inflow Γ i K outflow Γ o advection vector Jay Gopalakrishnan 14/46
32 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Jay Gopalakrishnan 15/46
33 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ i Γ i For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Jay Gopalakrishnan 15/46
34 Advection on tents Γ o The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ o For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Outflow boundary Γ o = part of K where n t + α n x > 0. Jay Gopalakrishnan 15/46
35 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Γ io For scalar u on tents K, define Âu = u d t + u α i. x i i=1 Inflow boundary Γ i = part of K where n t + α n x < 0. Outflow boundary Γ o = part of K where n t + α n x > 0. Define Γ io = Γ i Γ o and δ(z) = dist(z, Γ io ). Jay Gopalakrishnan 15/46
36 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.36 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
37 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.25 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
38 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.18 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
39 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0.1 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
40 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x =0 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
41 Advection on tents The known result does not apply to tents because dist(γ i, Γ o ) = 0. Γ io Assumption [T]: Tent pole height Assume that tent s bottom and top are contained in Γ i and Γ o, respectively. n t n x = 0.04 Top n n x n t Illustration: Â = t x and α = 1: On Γ o : n t + α n x = n t n x > 0. Top Γ o. Bottom Jay Gopalakrishnan 15/46
42 Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 For smooth functions w on a tent K, let τ i w = w Γi, τ o w = w Γo. Theorem [G+Monk+Sepúlveda 2015] Suppose Assumption [T] holds. Then, the maps τ i and τ o extend to continuous linear operators Moreover, are continuous. τ i : Ŵ L 2 δ (Γ i) and τ o : Ŵ L 2 δ (Γ o). τ i : ker(τ o ) L 2 1/δ (Γ i) and τ o : ker(τ i ) L 2 1/δ (Γ o) Jay Gopalakrishnan 16/46
43 Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 Interpreting τ i : Ŵ L 2 δ (Γ i) continuity: Inflow trace τ i w may behave badly near Γ io because we can only expect Γ i δ τ i w 2 <. Γ io Γ i Γ io δ 0 Jay Gopalakrishnan 16/46
44 Trace theorem for advection on tents Let Âu = u d t + u α i and let Ŵ be the graph space of Â. x i i=1 Interpreting τ i : ker(τ o ) L 2 1/δ (Γ i) continuity: If w Γo = 0, then to satisfy (τ i w)(x) 0 weakly as x Γ io, Γ i 1 δ τ iw 2 <. Γ io Γ i Γ io δ 0 Jay Gopalakrishnan 16/46
45 Summary of corollaries for the wave operator Traces. For any w W, the graph space of wave operator A, the trace w tent bottom exist in a weighted Lebesgue space. Define V = {w W : w tent bottom = 0}. Wellposedness. Friedrichs theory = A : V L 2 is a bijection. Our tent problem is uniquely solvable for any f L 2. { Au = f on K, u V, b.i.c. on K. Discretization. Continuous Lagrange finite element space satisfies the weak continuity requirements at Γ io. A stable test space generates a Petrov Galerkin discretization on tents. [G+Monk+Sepúlveda 2015] Jay Gopalakrishnan 17/46
46 Uniform grid error analysis On uniform grids, the method (after condensation) reduces to the leap frog scheme for systems. [ U n+1] j V n+1 j tent [ U n j 1 V n j 1 ] [ U n ] j+1 V n j+1 [ U n 1] j V n 1 j h/2 k/2 Von Neumann analysis shows second order convergence whenever Assumption [T] holds. Jay Gopalakrishnan 18/46
47 His 4 golden rules FUN NEW! DOABLE IMPORTANT Jay Gopalakrishnan 19/46
48 Next Introduction to tent pitching ! Basic idea ! History & background ! A one-dimensional example ! One-dimensional wave equation ! New mathematics for tent spaces ! General multidimensional hyperbolic systems Multidimensional tents Advantages & drawbacks Introducing MTP schemes Maps Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 20/46
49 The problem A general hyperbolic problem Let Ω 0 R N be a spatial domain. On the spacetime domain Ω = Ω 0 (0, T ), we are given smooth functions f : Ω R L R L N, g : Ω R L R L, b : Ω R L R L. Find a function u : Ω R L satisfying t g(x, t, u(x, t)) + div xf (x, t, u(x, t)) + b(x, t, u(x, t)) = 0 where div x is the spatial (x) divergence applied row-wise. Assume that this system of L equations is hyperbolic in the t-direction. Jay Gopalakrishnan 21/46
50 Tent meshing a time slab Solvable in parallel: Initial level 1 (red) tents Jay Gopalakrishnan 22/46
51 Tent meshing a time slab Solvable in parallel: Level 2 (orange) tents Jay Gopalakrishnan 22/46
52 Tent meshing a time slab Solvable in parallel: Level 3 (yellow) tents Jay Gopalakrishnan 22/46
53 Tent meshing a time slab Solvable in parallel: Level 4 tents Jay Gopalakrishnan 22/46
54 Tent meshing a time slab Solvable in parallel: Level 5 tents Jay Gopalakrishnan 22/46
55 Tent meshing a time slab Solvable in parallel: Level 6 tents (completing a time slab) Jay Gopalakrishnan 22/46
56 Tent meshing a time slab Solvable in parallel: Level 6 tents (completing a time slab) Next, repeatedly stack this mesh in time Jay Gopalakrishnan 22/46
57 Advantages A rational way to incorporate high order approximations, spatial adaptivity, and locally varying time steps, even on complex structures. Jay Gopalakrishnan 23/46
58 Advantages A rational way to incorporate high order approximations, spatial adaptivity, and locally varying time steps, even on complex structures. Tent pole height restriction is a local CFL constraint. In contrast, in standard timestepping, time step is constrained by the global CFL measure minimal mesh size (maximal degree) 1 2 wave speed Task parallelism is possible, even when widely varying time steps are used at different spatial locations. In contrast, standard timestepping requires delicate synchronization steps when using local time stepping. Jay Gopalakrishnan 23/46
59 How to improve? t K t All spacetime unknowns within a tent are coupled. x In traditional time stepping, only spatial unknowns are coupled. x The separation of time and space in traditional time stepping avoided excessive coupling of degrees of freedom. Jay Gopalakrishnan 24/46
60 A new twist: Maps Spacetime Tent Spacetime Cylinder Φ t K v ˆt ˆKˆKˆKˆKˆK v x v ˆx Instead of solving on tent K v, solve after pulling back to a domain where time and space discretizations can be easily separated. Jay Gopalakrishnan 25/46
61 Introducing MTP Schemes Mapped Tent Pitching (MTP) schemes have these steps: 1 Construct spacetime mesh using a tent pitching (meshing) algorithm. 2 Map conservation law on each tent to a spacetime cylinder. 3 Spatially discretize on the cylinder using your favorite method. 4 Apply explicit or implicit high order time stepping within the cylinder. 5 Map computed solution on the cylinder back to the tent. Jay Gopalakrishnan 26/46
62 Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: K v Φ ˆK v = Ω v (0, 1) t ˆt v Ω v Jay Gopalakrishnan 27/46
63 Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: If tent bottom is the graph of ϕ b, t v K v Ω v Φ ˆt ˆK v = Ω v (0, 1) Jay Gopalakrishnan 27/46
64 Form of the (Duffy-like) map The map is ) ( ) ˆx Φ = (ˆxˆt ϕ(ˆx, ˆt) where ϕ is defined as follows: If then tent bottom is the graph of ϕ b, tent top is the graph of ϕ t, ϕ = (1 ˆt)ϕ b + ˆtϕ t. t v K v Ω v Important weight: δ = ϕ t ϕ b. Φ ˆt ˆK v = Ω v (0, 1) Jay Gopalakrishnan 27/46
65 Pull back of the conservation law Let ˆf (w) = f (Φ(ˆx, ˆt), w), ĝ(w) = g(φ(ˆx, ˆt), w), ˆb(w) = b(φ(ˆx, ˆt), w), Ĝ(w) = ĝ(w) ˆf (w) gradˆx ϕ. (To simplify notation, suppress dependence of Ĝ, ˆf, ˆb on ˆx, ˆt.) Theorem [G+Schöberl+Wintersteiger 2016] A function u : K v R L satisfies the hyperbolic equation t g(u) + div xf (u) + b(u) = 0 in K v, if and only if û = u Φ satisfies, with δ = ϕ t ϕ b, ( Ĝ(û) + divˆx δ ˆf (û) ) + δ ˆb(û) = 0 in ˆK v. ˆt Idea of proof: Write the equation using the spacetime divergence operator div xt F + b = 0, apply the Piola map, and manipulate. Jay Gopalakrishnan 28/46
66 Next Introduction to tent pitching ! Basic idea ! History & background ! A one-dimensional example One-dimensional wave equation ! New mathematics for tent spaces ! General multidimensional hyperbolic systems Multidimensional tents ! Advantages & drawbacks ! Introducing MTP schemes ! Maps ! Examples: Wave, Maxwell & Euler systems Jay Gopalakrishnan 29/46
67 Wave equation Classical form of the wave problem Find φ : Ω R s.t. tt φ + β t φ div x (αgrad x φ) = 0 in Ω := Ω 0 (0, T ), n x αgrad x φ = 0 on Ω 0 (0, T ) φ = φ 0 on Ω 0 t φ = φ 1 on Ω 0 Jay Gopalakrishnan 30/46
68 Wave equation Classical form of the wave problem Find φ : Ω R s.t. tt φ + β t φ div x (αgrad x φ) = 0 in Ω := Ω 0 (0, T ), n x αgrad x φ = 0 on Ω 0 (0, T ) φ = φ 0 on Ω 0 t φ = φ 1 on Ω 0 [ ] [ ] q αgradx φ Reformulate as a first order system using = µ t φ [ ] [ ] [ ] [ ] [ ] [ ] α 1 0 q 0 gradx q 0 0 q t + = µ div x 0 µ 0 β µ }{{}}{{}}{{} g(x,t,u) div x f (u) b(x,t,u) : Jay Gopalakrishnan 30/46
69 An MTP scheme for the wave equation Equation to solve after mapping to cylinder (apply the theorem): [ ˆα 1 ] ] [ ] [ ] gradˆx ϕ [ˆq ˆt (gradˆx ϕ) t gradˆx (δˆµ) ˆµ divˆx (δˆq) δ ˆβ ˆµ }{{}}{{}}{{} Ĝ(û) divˆx δˆf (û) δˆb(û) = 0 Use the order p Raviart-Thomas pair of spaces (for ˆq and ˆµ) for the spatial (ˆx) discretization within the cylinder. Use a p-stage implicit Runge-Kutta method for time discretization within the cylinder. Jay Gopalakrishnan 31/46
70 Wave solution by locally implicit MTP scheme Use MTP scheme (β = 0) to approximate the standing wave φ = cos(πx 1 ) cos(πx 2 ) sin( πt 2 ) 2π. Implementation in NGSolve. Observed O(h p ) convergence. Jay Gopalakrishnan 32/46
71 His 4 golden rules FUN NEW DOABLE! IMPORTANT Jay Gopalakrishnan 33/46
72 Doable in 4D! Unified implementation for spatial DIM=2 and DIM=3 possible. 4D tents only need 3D spatial mesh data and tentpole heights. Jay Gopalakrishnan 34/46
73 3D Maxwell Resonator Spatial mesh is refined along sharp inner curves. MTP scheme automatically adapts local time steps. [Hochbruck+Paur+Schulz+Thawinan+Wieners 2015] Jay Gopalakrishnan 35/46
74 3D Maxwell Resonator Gaussian electric field pulse used to start simulation. 30 million dofs Hy component of the solution Jay Gopalakrishnan Simulated up to t=260 s with p = 2 in 20 min on E cores 35/46
75 His 4 golden rules FUN NEW DOABLE IMPORTANT! Jay Gopalakrishnan 36/46
76 For the first time, we can: 1 Perform explicit time stepping through unstructured tent meshes. 2 Use standard spatial discretizations within tents. Jay Gopalakrishnan 37/46
77 Data locality example Example: Scalar hyperbolic problem with p = 3 in 3 space dim. We have 20 spatial DG shape functions per spatial tetrahedron. Assume 20 simplices form a 4D spacetime tent. Time propagation operator acts on 400 unknowns. Matrix-free implementation of MTP schemes need only load into memory 400 double precision numbers, or KB. 3KB easily fits into local cache. = Timesteps within a tent require little memory movement. Jay Gopalakrishnan 38/46
78 Euler equations Euler equations modeling an ideal gas: N = 2, L = 5, with t g(u) + div xf (u) + b(u) = 0, ρ m u = m, g(u) = u, f (u) = PI + m m/ρ, b = 0, E (E + P)m/ρ where the pressure P is related to the state variables by P = 1 2 ρt, T = 4 ( E 5 ρ 1 m 2 ) 2 ρ 2. ρ : Ω R m : Ω R N E : Ω R density momentum total energy Jay Gopalakrishnan 39/46
79 Wind tunnel Well-known test problem: Mach 3 wind tunnel with a flat-faced step. [Emery 1968] [Woodward+Colella 1984] x 2 1 reflect inflow outflow 0.2 reflect reflect reflect x 1 At t = 0: ρ = 1.4, m = ρ [ 3 0 ] t, P = 1. Jay Gopalakrishnan 40/46
80 Historical computations i ++*. * c. * %. -.. * I * I I,--/ f *- l FIG. 11. Isobars (P/PJ for two-dimensional step as given by Lax-Wendroff method: o = 0.7, n = 1397, ndt = Ax/c1 (+ = 10, d = 11) (K was changed from 2.0 to 4.0 at nat = Ax/c,) ;t. 21 i -.. * * c-.a*, * z<n I. -. s-:n -c 8 -.* -* a * *--a [Emery 1968] l 5 l l. + r... a z *.. -.-fl.. I....I (L * * +,r..: -n....#., l.. l c.*.i.. - * -.* c..#. * ? l -.. c-s NC *. =; ;,; m...:*-* --, l -.- *. *. *... l l -0. *. -. m l.. *. -. l * * l I. c.-. * -w* I - --r * l * I ; l * -* * I. 1.*, * Jay Gopalakrishnan 41/46
81 Historical computations [Woodward+Colella 1984] FIG. 3-Continued. Jay Gopalakrishnan 41/46
82 Tent pitched time slab Jay Gopalakrishnan 42/46
83 Tent pitched time slab Jay Gopalakrishnan 42/46
84 Results MTP makes it possible to use standard spatial DG discretization on tents. t=4, degree 4 spatial DG, 3951 triangles [Video of time steps] Jay Gopalakrishnan 43/46
85 Ongoing work: Error analysis Analysis following the technique of [Monk+Richter 2005]: Split spacetime mesh into layers of tents. Analyze error propagation within each layer. Reduce the problem to analysis of the error in a single tent. Jay Gopalakrishnan 44/46
86 Ongoing work: Improved timestepping Tent system to solve after mapping: Reduction to ODE: Put V = MU. Then ( Ĝ(û) + divˆx δ ˆf (û) ) + δ ˆb(û) = 0 in ˆK v. ˆt d dˆt [MU](ˆt) + AU(ˆt) = 0 d dˆt V + AM 1 V = 0 The presence of M 1 prevents straightforward error analysis. Taylor time stepping preserves polynomials, so holds promise. Jay Gopalakrishnan 45/46
87 Conclusion New mathematics needed for understanding tent spaces. [G+Monk+Sepúlveda 2015] A tent pitching scheme motivated by Friedrichs theory MTP schemes provide a spacetime discretization that is high order in space and time, is constrained by a local (not global) CFL condition, has new efficiencies due to mapping and decoupling, gives explicit schemes on unstructured spacetime meshes, allows the use of standard DG methods on tents. [G+Schöberl+Wintersteiger 2017] Mapped tent pitching schemes for hyperbolic systems Happy birthday Bernardo! Jay Gopalakrishnan 46/46
Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of. Conservation Laws 1. Abstract
Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Conservation Laws Sirui Tan and Chi-Wang Shu 3 Abstract We develop a high order finite difference numerical boundary condition for solving
More informationA space-time Trefftz method for the second order wave equation
A space-time Trefftz method for the second order wave equation Lehel Banjai The Maxwell Institute for Mathematical Sciences Heriot-Watt University, Edinburgh Rome, 10th Apr 2017 Joint work with: Emmanuil
More information7 Hyperbolic Differential Equations
Numerical Analysis of Differential Equations 243 7 Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena.
More informationPDEs, part 3: Hyperbolic PDEs
PDEs, part 3: Hyperbolic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2011 Hyperbolic equations (Sections 6.4 and 6.5 of Strang). Consider the model problem (the
More informationAProofoftheStabilityoftheSpectral Difference Method For All Orders of Accuracy
AProofoftheStabilityoftheSpectral Difference Method For All Orders of Accuracy Antony Jameson 1 1 Thomas V. Jones Professor of Engineering Department of Aeronautics and Astronautics Stanford University
More information30 crete maximum principle, which all imply the bound-preserving property. But most
3 4 7 8 9 3 4 7 A HIGH ORDER ACCURATE BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME FOR SCALAR CONVECTION DIFFUSION EQUATIONS HAO LI, SHUSEN XIE, AND XIANGXIONG ZHANG Abstract We show that the classical
More informationA Stable Spectral Difference Method for Triangles
A Stable Spectral Difference Method for Triangles Aravind Balan 1, Georg May 1, and Joachim Schöberl 2 1 AICES Graduate School, RWTH Aachen, Germany 2 Institute for Analysis and Scientific Computing, Vienna
More informationA space-time Trefftz method for the second order wave equation
A space-time Trefftz method for the second order wave equation Lehel Banjai The Maxwell Institute for Mathematical Sciences Heriot-Watt University, Edinburgh & Department of Mathematics, University of
More informationFinite Element methods for hyperbolic systems
Finite Element methods for hyperbolic systems Eric Sonnendrücker Max-Planck-Institut für Plasmaphysik und Zentrum Mathematik, TU München Lecture notes Wintersemester 14-15 January 19, 15 Contents 1 Introduction
More informationDivergence Formulation of Source Term
Preprint accepted for publication in Journal of Computational Physics, 2012 http://dx.doi.org/10.1016/j.jcp.2012.05.032 Divergence Formulation of Source Term Hiroaki Nishikawa National Institute of Aerospace,
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationA spacetime DPG method for acoustic waves
A spacetime DPG method for acoustic waves Rainer Schneckenleitner JKU Linz January 30, 2018 ( JKU Linz ) Spacetime DPG method January 30, 2018 1 / 41 Overview 1 The transient wave problem 2 The broken
More informationWeak form of Boundary Value Problems. Simulation Methods in Acoustics
Weak form of Boundary Value Problems Simulation Methods in Acoustics Note on finite dimensional description of functions Approximation: N f (x) ˆf (x) = q j φ j (x) j=1 Residual function: r(x) = f (x)
More informationPerformance comparison between hybridizable DG and classical DG methods for elastic waves simulation in harmonic domain
March 4-5, 2015 Performance comparison between hybridizable DG and classical DG methods for elastic waves simulation in harmonic domain M. Bonnasse-Gahot 1,2, H. Calandra 3, J. Diaz 1 and S. Lanteri 2
More informationA local-structure-preserving local discontinuous Galerkin method for the Laplace equation
A local-structure-preserving local discontinuous Galerkin method for the Laplace equation Fengyan Li 1 and Chi-Wang Shu 2 Abstract In this paper, we present a local-structure-preserving local discontinuous
More informationENO and WENO schemes. Further topics and time Integration
ENO and WENO schemes. Further topics and time Integration Tefa Kaisara CASA Seminar 29 November, 2006 Outline 1 Short review ENO/WENO 2 Further topics Subcell resolution Other building blocks 3 Time Integration
More informationA Hybrid Method for the Wave Equation. beilina
A Hybrid Method for the Wave Equation http://www.math.unibas.ch/ beilina 1 The mathematical model The model problem is the wave equation 2 u t 2 = (a 2 u) + f, x Ω R 3, t > 0, (1) u(x, 0) = 0, x Ω, (2)
More informationBasic Aspects of Discretization
Basic Aspects of Discretization Solution Methods Singularity Methods Panel method and VLM Simple, very powerful, can be used on PC Nonlinear flow effects were excluded Direct numerical Methods (Field Methods)
More informationNew DPG techniques for designing numerical schemes
New DPG techniques for designing numerical schemes Jay Gopalakrishnan University of Florida Collaborator: Leszek Demkowicz October 2009 Massachusetts Institute of Technology, Boston Thanks: NSF Jay Gopalakrishnan
More informationStable and high-order accurate finite difference schemes on singular grids
Center for Turbulence Research Annual Research Briefs 006 197 Stable and high-order accurate finite difference schemes on singular grids By M. Svärd AND E. van der Weide 1. Motivation and objectives The
More informationStrong Stability Preserving Time Discretizations
AJ80 Strong Stability Preserving Time Discretizations Sigal Gottlieb University of Massachusetts Dartmouth Center for Scientific Computing and Visualization Research November 20, 2014 November 20, 2014
More informationDesign of optimal Runge-Kutta methods
Design of optimal Runge-Kutta methods David I. Ketcheson King Abdullah University of Science & Technology (KAUST) D. Ketcheson (KAUST) 1 / 36 Acknowledgments Some parts of this are joint work with: Aron
More informationDiscontinuous Galerkin Methods
Discontinuous Galerkin Methods Joachim Schöberl May 20, 206 Discontinuous Galerkin (DG) methods approximate the solution with piecewise functions (polynomials), which are discontinuous across element interfaces.
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 25: Introduction to Discontinuous Galerkin Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods
More informationChapter 1. Introduction and Background. 1.1 Introduction
Chapter 1 Introduction and Background 1.1 Introduction Over the past several years the numerical approximation of partial differential equations (PDEs) has made important progress because of the rapid
More informationBasics on Numerical Methods for Hyperbolic Equations
Basics on Numerical Methods for Hyperbolic Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro October 8,
More informationTutorial 2. Introduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2
More informationPart 1. The diffusion equation
Differential Equations FMNN10 Graded Project #3 c G Söderlind 2016 2017 Published 2017-11-27. Instruction in computer lab 2017-11-30/2017-12-06/07. Project due date: Monday 2017-12-11 at 12:00:00. Goals.
More informationNumerical Methods for PDEs
Numerical Methods for PDEs Problems 1. Numerical Differentiation. Find the best approximation to the second drivative d 2 f(x)/dx 2 at x = x you can of a function f(x) using (a) the Taylor series approach
More informationAdvection / Hyperbolic PDEs. PHY 604: Computational Methods in Physics and Astrophysics II
Advection / Hyperbolic PDEs Notes In addition to the slides and code examples, my notes on PDEs with the finite-volume method are up online: https://github.com/open-astrophysics-bookshelf/numerical_exercises
More informationThe Discontinuous Galerkin Method for Hyperbolic Problems
Chapter 2 The Discontinuous Galerkin Method for Hyperbolic Problems In this chapter we shall specify the types of problems we consider, introduce most of our notation, and recall some theory on the DG
More information2 Multidimensional Hyperbolic Problemss where A(u) =f u (u) B(u) =g u (u): (7.1.1c) Multidimensional nite dierence schemes can be simple extensions of
Chapter 7 Multidimensional Hyperbolic Problems 7.1 Split and Unsplit Dierence Methods Our study of multidimensional parabolic problems in Chapter 5 has laid most of the groundwork for our present task
More informationLocal discontinuous Galerkin methods for elliptic problems
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2002; 18:69 75 [Version: 2000/03/22 v1.0] Local discontinuous Galerkin methods for elliptic problems P. Castillo 1 B. Cockburn
More informationA Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations
A Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations Ch. Altmann, G. Gassner, F. Lörcher, C.-D. Munz Numerical Flow Models for Controlled
More informationConservation Laws & Applications
Rocky Mountain Mathematics Consortium Summer School Conservation Laws & Applications Lecture V: Discontinuous Galerkin Methods James A. Rossmanith Department of Mathematics University of Wisconsin Madison
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationMath 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework
Math 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework Jan Mandel University of Colorado Denver May 12, 2010 1/20/09: Sec. 1.1, 1.2. Hw 1 due 1/27: problems
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear
More informationPositivity-preserving high order schemes for convection dominated equations
Positivity-preserving high order schemes for convection dominated equations Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Xiangxiong Zhang; Yinhua Xia; Yulong Xing; Cheng
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationSuperconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients
Superconvergence of discontinuous Galerkin methods for -D linear hyperbolic equations with degenerate variable coefficients Waixiang Cao Chi-Wang Shu Zhimin Zhang Abstract In this paper, we study the superconvergence
More informationPAijpam.eu NEW H 1 (Ω) CONFORMING FINITE ELEMENTS ON HEXAHEDRA
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 609-617 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v109i3.10
More informationarxiv: v1 [math.na] 22 Nov 2018
Asymptotic preserving Deferred Correction Residual Distribution schemes Rémi Abgrall and Davide Torlo arxiv:1811.09284v1 [math.na] 22 Nov 2018 Abstract This work aims to extend the residual distribution
More informationOn the efficiency of the Peaceman-Rachford ADI-dG method for wave-type methods
On the efficiency of the Peaceman-Rachford ADI-dG method for wave-type methods Marlis Hochbruck, Jonas Köhler CRC Preprint 2017/34 (revised), March 2018 KARLSRUHE INSTITUTE OF TECHNOLOGY KIT The Research
More informationCS205B / CME306 Homework 3. R n+1 = R n + tω R n. (b) Show that the updated rotation matrix computed from this update is not orthogonal.
CS205B / CME306 Homework 3 Rotation Matrices 1. The ODE that describes rigid body evolution is given by R = ω R. (a) Write down the forward Euler update for this equation. R n+1 = R n + tω R n (b) Show
More informationApplying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models
0-0 Applying Asymptotic Approximations to the Full Two-Fluid Plasma System to Study Reduced Fluid Models B. Srinivasan, U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle,
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationNumerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement
Numerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement Romain Teyssier CEA Saclay Romain Teyssier 1 Outline - Euler equations, MHD, waves, hyperbolic
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationChapter 1. Introduction
Chapter 1 Introduction Many astrophysical scenarios are modeled using the field equations of fluid dynamics. Fluids are generally challenging systems to describe analytically, as they form a nonlinear
More informationThe Discontinuous Galerkin Method on Cartesian Grids with Embedded Geometries: Spectrum Analysis and Implementation for Euler Equations
The Discontinuous Galerkin Method on Cartesian Grids with Embedded Geometries: Spectrum Analysis and Implementation for Euler Equations by Ruibin Qin A thesis presented to the University of Waterloo in
More informationImplicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method
Implicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method Per-Olof Persson and Jaime Peraire Massachusetts Institute of Technology 7th World Congress on Computational Mechanics
More informationHIGH ORDER LAX-WENDROFF-TYPE SCHEMES FOR LINEAR WAVE PROPAGATION
European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006 P. Wesseling, E. Oñate and J. Périaux (Eds) c TU Delft, The Netherlands, 2006 HIGH ORDER LAX-WENDROFF-TYPE SCHEMES FOR LINEAR WAVE PROPAGATION
More informationHp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme
Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme Aravind Balan, Michael Woopen and Georg May AICES Graduate School, RWTH Aachen University, Germany 22nd AIAA Computational
More informationOn limiting for higher order discontinuous Galerkin method for 2D Euler equations
On limiting for higher order discontinuous Galerkin method for 2D Euler equations Juan Pablo Gallego-Valencia, Christian Klingenberg, Praveen Chandrashekar October 6, 205 Abstract We present an implementation
More informationA High-Order Discontinuous Galerkin Method for the Unsteady Incompressible Navier-Stokes Equations
A High-Order Discontinuous Galerkin Method for the Unsteady Incompressible Navier-Stokes Equations Khosro Shahbazi 1, Paul F. Fischer 2 and C. Ross Ethier 1 1 University of Toronto and 2 Argonne National
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationMixed Finite Element Methods. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016
Mixed Finite Element Methods Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016 Linear elasticity Given the load f : Ω R n, find the displacement u : Ω R n and the
More informationFINITE DIFFERENCE APPROXIMATIONS FOR THE FIRST-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH POINT-WISE DELAY
International Journal of Pure and Applied Mathematics Volume 67 No. 2, 49-67 FINITE DIFFERENCE APPROXIMATIONS FOR THE FIRST-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH POINT-WISE DELAY Parameet
More informationNUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer:
Prof. Dr. O. Junge, T. März Scientific Computing - M3 Center for Mathematical Sciences Munich University of Technology Name: NUMERICAL ANALYSIS 2 - FINAL EXAM Summer Term 2006 Matrikelnummer: I agree to
More informationVector and scalar penalty-projection methods
Numerical Flow Models for Controlled Fusion - April 2007 Vector and scalar penalty-projection methods for incompressible and variable density flows Philippe Angot Université de Provence, LATP - Marseille
More informationGALERKIN TIME STEPPING METHODS FOR NONLINEAR PARABOLIC EQUATIONS
GALERKIN TIME STEPPING METHODS FOR NONLINEAR PARABOLIC EQUATIONS GEORGIOS AKRIVIS AND CHARALAMBOS MAKRIDAKIS Abstract. We consider discontinuous as well as continuous Galerkin methods for the time discretization
More informationFinite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
More informationEfficient wave propagation on complex domains
Center for Turbulence Research Annual Research Briefs 2006 223 Efficient wave propagation on complex domains By K. Mattsson, F. Ham AND G. Iaccarino 1. Motivation and objectives In many applications, such
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 59 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS The Finite Volume Method These slides are partially based on the recommended textbook: Culbert B.
More informationBound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu
Bound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu Division of Applied Mathematics Brown University Outline Introduction Maximum-principle-preserving for scalar conservation
More informationA Fifth Order Flux Implicit WENO Method
A Fifth Order Flux Implicit WENO Method Sigal Gottlieb and Julia S. Mullen and Steven J. Ruuth April 3, 25 Keywords: implicit, weighted essentially non-oscillatory, time-discretizations. Abstract The weighted
More informationA High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations
Motivation Numerical methods Numerical tests Conclusions A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations Xiaofeng Cai Department of Mathematics
More informationHigh-Order Finite-Volume Methods! Phillip Colella! Computational Research Division! Lawrence Berkeley National Laboratory!
High-Order Finite-Volume Methods! Phillip Colella! Computational Research Division! Lawrence Berkeley National Laboratory! Why Higher Order?! Locally-refined grids, mapped-multiblock grids smooth except
More informationShock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells
Abstract We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in sub-cells and treated by finite volume techniques Hence, our approach
More informationHybrid (DG) Methods for the Helmholtz Equation
Hybrid (DG) Methods for the Helmholtz Equation Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University of Technology Contributions by
More informationHigh Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation
High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation Faheem Ahmed, Fareed Ahmed, Yongheng Guo, Yong Yang Abstract This paper deals with
More informationA Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations
A Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations Hao Li Math Dept, Purdue Univeristy Ocean University of China, December, 2017 Joint work with
More informationNumerical Analysis Preliminary Exam 10.00am 1.00pm, January 19, 2018
Numerical Analysis Preliminary Exam 0.00am.00pm, January 9, 208 Instructions. You have three hours to complete this exam. Submit solutions to four (and no more) of the following six problems. Please start
More informationGlobal Maxwellians over All Space and Their Relation to Conserved Quantites of Classical Kinetic Equations
Global Maxwellians over All Space and Their Relation to Conserved Quantites of Classical Kinetic Equations C. David Levermore Department of Mathematics and Institute for Physical Science and Technology
More informationA parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows.
A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows Tao Xiong Jing-ei Qiu Zhengfu Xu 3 Abstract In Xu [] a class of
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationA minimum entropy principle of high order schemes for gas dynamics. equations 1. Abstract
A minimum entropy principle of high order schemes for gas dynamics equations iangxiong Zhang and Chi-Wang Shu 3 Abstract The entropy solutions of the compressible Euler equations satisfy a minimum principle
More informationSpectral element schemes for the. Korteweg-de Vries and Saint-Venant equations
Spectral element schemes for the Korteweg-de Vries and Saint-Venant equations R. PASQUETTI a a. Université Côte d Azur, CNRS, Inria, LJAD, France, richard.pasquetti@unice.fr... Résumé : Les sytèmes hyperboliques
More informationOn the Comparison of the Finite Volume and Discontinuous Galerkin Methods
Diploma Thesis Institute for Numerical Simulation, TUHH On the Comparison of the Finite Volume and Discontinuous Galerkin Methods Corrected version Katja Baumbach August 17, 2006 Supervisor: Prof. Dr.
More informationNumerical Analysis of Differential Equations Numerical Solution of Parabolic Equations
Numerical Analysis of Differential Equations 215 6 Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012 Numerical Analysis of Differential
More informationWeak Galerkin Finite Element Scheme and Its Applications
Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015 Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer
More informationStable Numerical Scheme for the Magnetic Induction Equation with Hall Effect
Stable Numerical Scheme for the Magnetic Induction Equation with Hall Effect Paolo Corti joint work with Siddhartha Mishra ETH Zurich, Seminar for Applied Mathematics 17-19th August 2011, Pro*Doc Retreat,
More information2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
More informationarxiv: v1 [math.na] 11 Jun 2018
High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics arxiv:1806.03986v1 [math.na] 11 Jun 2018 R. Abgrall, P. Bacigaluppi, S. Tokareva Institute of Mathematics,
More information1. Introduction. We consider the model problem that seeks an unknown function u = u(x) satisfying
A SIMPLE FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC PROBLEMS LIN MU AND XIU YE Abstract. In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy
More informationFEM Convergence for PDEs with Point Sources in 2-D and 3-D
FEM Convergence for PDEs with Point Sources in 2-D and 3-D Kourosh M. Kalayeh 1, Jonathan S. Graf 2 Matthias K. Gobbert 2 1 Department of Mechanical Engineering 2 Department of Mathematics and Statistics
More informationA positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations
A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations Sashank Srinivasan a, Jonathan Poggie a, Xiangxiong Zhang b, a School of Aeronautics and Astronautics,
More informationContinuous adjoint based error estimation and r-refinement for the active-flux method
Continuous adjoint based error estimation and r-refinement for the active-flux method Kaihua Ding, Krzysztof J. Fidkowski and Philip L. Roe Department of Aerospace Engineering, University of Michigan,
More informationScalable Non-Linear Compact Schemes
Scalable Non-Linear Compact Schemes Debojyoti Ghosh Emil M. Constantinescu Jed Brown Mathematics Computer Science Argonne National Laboratory International Conference on Spectral and High Order Methods
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More informationA Robust CFL Condition for the Discontinuous Galerkin Method on Triangular Meshes
A Robust CFL Condition for the Discontinuous Galerkin Method on Triangular Meshes N. Chalmers and L. Krivodonova Abstract When the discontinuous Galerkin (DG) method is applied to hyperbolic problems in
More informationHybridized DG methods
Hybridized DG methods University of Florida (Banff International Research Station, November 2007.) Collaborators: Bernardo Cockburn University of Minnesota Raytcho Lazarov Texas A&M University Thanks:
More informationScientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More informationKrylov single-step implicit integration factor WENO methods for advection-diffusion-reaction equations
Accepted Manuscript Krylov single-step implicit integration factor WENO methods for advection diffusion reaction equations Tian Jiang, Yong-Tao Zhang PII: S0021-9991(16)00029-2 DOI: http://dx.doi.org/10.1016/j.jcp.2016.01.021
More informationarxiv: v1 [math.na] 21 Nov 2017
High Order Finite Difference Schemes for the Heat Equation Whose Convergence Rates are Higher Than Their Truncation Errors, A. Ditkowski arxiv:7.0796v [math.na] Nov 07 Abstract Typically when a semi-discrete
More information