Journal ArticleDOI

# A Cartesian grid embedded boundary method for the heat equation on irregular domains

13 Nov 2001-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 173, Iss: 2, pp 620-635

TL;DR: An algorithm for solving the heat equation on irregular time-dependent domains is presented, based on the Cartesian grid embedded boundary algorithm of Johansen and Colella, combined with a second-order accurate discretization of the time derivative.

AbstractWe present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60--85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank--Nicolson time discretization is unstable, requiring us to use the L{sub 0}-stable implicit Runge--Kutta method of Twizell, Gumel, and Arigu.

Topics: , Boundary (topology) (59%), Discretization (56%), Heat equation (54%), Time derivative (52%)

## Summary (1 min read)

### 1. INTRODUCTION

• For the time discretization, for the fixed-boundary problem the authors use either the Crank-Nicolson method or the method of Twizell, Gumel and Arigu (TGA) [10] .
• The authors algorithm is stable and achieves second-order accuracy both on problems with fixed domain and on problems with a time-dependent domain (t) with boundaries moving with constant velocities.

### FIG. 1.

• Centers of cells in (t old ) are shown with solid circles, and centers of cells in (tnew) -(t old ) are shown with unfilled circles.
• The authors solve (19) numerically on a rectangular domain with three elliptically-shaped holes, with boundary conditions computed using the exact solution (18).
• In the moving-boundary problem, the holes move with constant velocities.
• With both fixed and moving boundaries, the authors solve two separate problems with different boundary conditions: Dirichlet conditions on all boundaries; Dirichlet conditions on the fixed external boundaries, but Neumann conditions on the boundaries of the ellipses.

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A Cartesian Grid Embedded Boundary Method for
the Heat Equation on Irregular Domains
1
Peter McCorquodale
y
, Phillip Colella
y
and Hans Johansen
z
y
Applied Numerical Algorithms Group, Lawrence Berkeley National Laboratory, Berkeley,
California 94720,
z
Department of Mechanical Engineering, University of California, Berkeley, California 94720
E-mail: PWMcCorquodale@lbl.gov
We presentan algorithm for solving the heatequationon irregular time-dependent
domains. It is basedon theCartesiangrid embeddedboundaryalgorithm of Johansen
and Colella (J. Comput. Phys. 147(2):60–85) for discretizing Poisson’s equation,
combined with a second-order accurate discretization of the time derivative. This
leadsto a method that is second-order accuratein spaceand time. For the case where
the boundary is moving, we convert the moving-boundary problem to a sequence
of ﬁxed-boundary problems, combined with an extrapolation procedure to initialize
values that are uncovered as the boundary moves. We ﬁnd that, in the moving
boundary case, the use of Crank–Nicolson time discretization is unstable, requiring
us to use the
L
0
-stable implicit Runge–Kuttamethod of Twizell, Gumel, andArigu.
Key Words: 35K15 Initial value problems for second-order, parabolic equations; embedded
boundary; moving boundaries.
1
Research supported at U.C. Berkeley by the U.S. Department of Energy Mathematical, Information and
Computing Sciences Division, Grants DE-FG03-94ER25205 and DE-FG03-92ER25140, and by the National
Science Foundation Graduate Fellowship Program; and at the Lawrence Berkeley National Laboratory by the
DRAFT March 14, 2001, 2:24pm DRAFT

2 MCCORQUODALE, COLELLA, JOHANSEN
1. INTRODUCTION
In this paper we present a numerical method for solving the parabolic initial-value
problem
t
=
D
+
f
on
,
(
x
;
0) =
0
(
x
)
(1)
with constant
D>
0
on a bounded region
, and boundary conditions of either Neumann
type
@
@n
=
g
n
(
x
;t
)
on
@
(2)
or Dirichlet type
=
g
d
(
x
;t
)
on
@
. (3)
As in previous work on elliptic problems [6], our approach uses a ﬁnite-volume dis-
cretization which embeds the domain in a regular Cartesian grid. We treat the solution as
cell-centered on a rectangular grid, even when the cell centers are outside the domain.
For the time discretization, for the ﬁxed-boundary problem we use either the Crank–
Nicolson method or the method of Twizell, Gumel and Arigu (TGA) [10]. We solve the
moving-boundaryproblem by converting it to a sequence of ﬁxed-boundary problems, and
applying the TGA method to each. Our algorithm is stable and achieves second-order
accuracy both on problems with ﬁxed domain
and on problems with a time-dependent
domain
(
t
)
with boundaries moving with constant velocities. If the ratio of timestep
t
to mesh spacing
h
is kept constant, then the solution error is
O
(
t
2
+
h
2
)
as
h;
t
!
0
.
Part of this work appeared in prelminary form in [7].
U.S. Department of Energy Mathematical, Information and Computing Sciences Division, Contract DE-AC03-
76SF00098.
DRAFT March 14, 2001, 2:24pm DRAFT

EMBEDDED BOUNDARY METHOD FOR THE HEAT EQUATION 3
2. THE HEAT EQUATION FOR FIXED BOUNDARIES
2.1. Spatial discretization
The underlying discretization of space is given by rectangular control volumes on a
Cartesian grid:
i
=[(
i
;
1
2
u
)
h;
(
i
+
1
2
u
)
h
]
,
i
2
Z
d
, where
d
is the dimensionalityof the
problem,
h
is the mesh spacing, and
u
is the vector whose entries are all ones. In the case
of a ﬁxed, irregular domain
, the geometry is represented by the intersection of
with
the Cartesian grid. We obtain control volumes
V
i
=
i
\
and faces
A
i
1
2
e
s
, that are
the intersection of
@V
i
with the coordinate planes
f
x
:
x
s
=(
i
s
1
2
)
h
g
. Here
e
s
is the
unit vector in the
s
direction. We also deﬁne
A
B
i
to be the intersection of the boundary of
the irregular domain with the Cartesian control volume:
A
B
i
=
@
\
i
. We will assume
here that there is a one-to-one correspondence between the control volumes and faces and
the corresponding geometric entities on the underlying Cartesian grid. The description can
be generalized to allow for boundaries whose width is less than the mesh spacing, or sharp
trailing edges.
In order to construct ﬁnite difference methods, we will need only a small number of
real-valued quantities that are derived from these geometric objects.
The areas / volumes, expressed in dimensionless terms: volume fractions
i
=
j
V
i
j
h
;
d
, face apertures
i
+
1
2
e
s
=
j
A
i
+
1
2
e
s
j
h
;
(
d
;
1)
and boundary apertures
B
i
=
j
A
B
i
j
h
;
(
d
;
1)
. We assume that we can compute estimates of the dimensionless quanti-
ties that are accurate to
O
(
h
2
)
.
The locations of centroids, and the average outward normal to the boundary.
x
i
=
1
j
V
i
j
Z
V
i
x
dV
x
i
+
1
2
e
s
=
1
j
A
i
+
1
2
e
s
j
Z
A
i
+
1
2
e
s
x
dA
x
B
i
=
1
j
A
B
i
j
Z
A
B
i
x
dA
DRAFT March 14, 2001, 2:24pm DRAFT

4 MCCORQUODALE, COLELLA, JOHANSEN
n
B
i
=
1
j
A
B
i
j
Z
A
B
i
n
B
dA
where
n
B
is the outward normal to
@
, deﬁned for each point on
@
. Again, we assume
that we can compute estimates of these quantities that are accurate to
O
(
h
2
)
.
Using just these quantities, we can deﬁne conservative discretizations for the divergence
operator. Let
~
F
=(
F
1
:::F
d
)
be a function of
x
. Then
r
~
F
1
j
V
i
j
Z
V
i
r
~
FdV
=
1
j
V
i
j
Z
@V
i
~
F
n
dA
1
i
h
(
X
=+
;
;
d
X
s
=1
i
1
2
e
s
F
s
(
x
i
1
2
e
s
)+
B
i
n
B
i
~
F
(
x
B
i
))
(4)
where (4) is obtained by replacing the integrals of the normal components of the vector
ﬁeld
~
F
with the values at the centroids.
We can use this idea to discretize the Laplacian, written as the divergence of a ﬂux:
=
r
~
F
where
~
F
=
r
. We follow the approach described in [6, 7]. The discretized
solution values approximate the solution to the PDE at the rectangular cell centers:
U
n
i
(
i
h; n
t
)
. At ﬁrst glance, this might be a cause for concern, since some of the centers
of Cartesian cells
i
might not be contained in
. However, it is well known that, for
any domain with smooth boundary, a smooth function can be extended to all of
R
d
with
a bound on the relative increase in the
C
k;
norms that depends only on the domain and
(
k;
)
[5]. We assume that the values
U
i
on the covered cell centers approximate such an
extension. We deﬁne the time-dependent inhomogeneous operator
L
h
I
(
t
)
(
L
h
I
(
t
)
U
)
i
=
1
i
h
(
X
=+
;
;
d
X
s
=1
i
1
2
e
s
F
s
i
1
2
e
s
+
B
i
n
B
i
~
F
(
x
B
i
;t
))
:
(5)
The ﬂuxes on the cell faces are computed from
U
by linearly interpolating between
centered difference approximations. For example, for the ﬁrst component (
s
=1
) in two
dimensions,
F
1
i
+
1
2
;j
=
(
U
i
+1
;j
;
U
i;j
)
h
+(1
;
)
(
U
i
+1
;j
1
;
U
i;j
1
)
h
(6)
DRAFT March 14, 2001, 2:24pm DRAFT

EMBEDDED BOUNDARY METHOD FOR THE HEAT EQUATION 5
=
j
y
i
+
1
2
;j
;
jh
j
h
(7)
where
=+(
;
)
if
y
i
+
1
2
;j
>jh
(
<jh
)
.
Since
~
F
=
r
, then
n
B
~
F
B
=
@
@n
(8)
and so with Neumann boundary conditions (2), we set
n
B
i
~
F
(
x
B
i
;t
)=
g
n
(
x
B
i
;t
)
in
(5). With Dirichlet boundary conditions as from (3), we compute an estimate of
@
@n
by
interpolating from the grid values and the values at the boundaries; for details, see [6].
For both Dirichlet and Neumann boundary conditions, these discretizations lead to linear
systems with the same asymptotic conditioning properties as those of the corresponding
operators in the absence of irregularboundaries, and are amenable tothe use of fast iterative
solvers such as multigrid. Finally, we denote by
L
h
H
the operator
L
h
I
(
t
)
with homogeneous
boundary conditions,
g
n
=0
or
g
d
=0
.
2.2. TGA temporal discretization
We apply themethod of Twizell, Gumel and Arigu [10] to solve the initial-valueproblem
dU
dt
=
L
h
I
(
t
)
U
(
t
)+
f
(
t
)
(9)
U
(0) =
U
0
where
f
is evaluated at the same cell centers as
U
.
We split the timestep
t
such that
1
+
2
+
3
=
t
1
+
2
+
4
=
t=
2
:
The update at step
n
uses the boundary values at the old and new times and also at an
intermediate time
t
int
:
U
n
+1
=(
I
;
1
L
h
I
(
t
new
))
;
1
(
I
;
2
L
h
I
(
t
int
))
;
1
DRAFT March 14, 2001, 2:24pm DRAFT

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##### References
More filters

Journal ArticleDOI

36,037 citations

• ...Similar approaches based on formally inconsistent discretizations at the irregular boundary have been used previously and observed to be stable [1, 9], so we expect that the extension to the more accurate boundary discretization should be straightforward....

[...]

Book
07 Jan 2013
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for

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Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by $$Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u$$ is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q, $$Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right)$$ [D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.

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442 citations

### "A Cartesian grid embedded boundary ..." refers background or methods in this paper

• ...As in previous work on elliptic problems [6], our approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid....

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• ...We follow the approach described in [6, 7]....

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• ...With Dirichlet boundary conditions as from (3), we compute an estimate of ∂ψ ∂n by interpolating from the grid values and the values at the boundaries; for details, see [6]....

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• ...The method described here, together with that in [6] for elliptic PDEs and [8] for hyperbolic PDEs, provide the fundamental components required for developing second-order accurate methods for a broad range of continuum mechanics problems in irregular geometries based on the predictor–corrector approach in [2]....

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• ...This is routine for the case in which the embedded boundary is contained in the finest level of refinement [6], but requires some additional discretization design when the embedded boundary crosses coarse–fine interfaces....

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##### Frequently Asked Questions (1)
###### Q1. What are the contributions mentioned in the paper "A cartesian grid embedded boundary method for the heat equation on irregular domains" ?

The authors present an algorithm for solving the heat equation on irregular time-dependent domains.