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1 Lecture 24: Flux Limiters
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2 Last Time… l Developed a set of limiter functions l Second order accurate
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3 This Time… l Examine physical rationale for limiter functions l Application to unstructured meshes
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4 Recall Higher-Order Scheme for e l Consider finding face value using a second-order scheme with the gradient found at the upwind cell: l Recall: l What is the limiter function trying to do?
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5 Limiter Functions =2r
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6 Physical Interpretation l The value of r can be thought of as the ratio of two gradients: l Limiter chooses gradient adaptively to avoid creating extrema Downwind cell gradient Upwind cell gradient ww
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7 Case (a): Linear Variation l Since: l If variation is a straight line, on a uniform mesh, r=1 l From our limiter function range, =1 for r=1 l Can use either gradient and get the right value at e r=1
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8 Case (b): 2>r>1 l r>1 means l If we used =1, we would not create overshoot l In fact we can use up to r and not create
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9 Case (b): 2>r>1 (Cont’d) l Consider case when r e >1, i.e., l Say we choose the =r e line l When =r e :
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10 Case (b’): r>2 l Consider case when r e >2, i.e., l For r e >2, say we choose the =2 line l When =2:
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11 Case (c): 0< r<1 l If r<1:
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12 Case (c): 0<r<1 (Cont’d) l Consider case when 0<r e <1, i.e., l Say we choose =r e l When =r e :
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13 Case (d): r<0 l When r<0, this implies local extremum l Our limiter has =0 for r<0 l This implies Defaults to first order upwind scheme
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14 Unstructured Meshes l Find face value using: l No easy way to define r f
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15 Unstructured Meshes Create fictitious point U Find value at U by using cell gradient Hence define r f
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16 Closure In this lecture, we l Considered the physical meaning of the limiter function l Saw that it was an adaptive way to choose either an upwind or a downwind gradient to find face value l Looked at difficulties in implementing for unstructured meshes
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