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PdeFiniteDifference.cu
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PdeFiniteDifference.cu
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#include "PdeFiniteDifference.cuh"
#include <CuSparseWrappers.cuh>
namespace detail
{
/**
* Evolve the solution using the time discretizer.
* N.B.: solution is a memory tile, as some solver might require the solution history
* N.B.2: if provided, workBuffer is a previously allocated buffer used for matrix-vector multiplication
*/
int _Advance(MemoryTile& solution, const MemoryCube& timeDiscretizer, MemoryTile& workBuffer, const bool overwriteBuffer)
{
// this is to support multi-step algorithms: each solution is multiplied by a different time discretizer
MemoryBuffer _solution(solution.pointer, solution.nRows, solution.memorySpace, solution.mathDomain);
MemoryBuffer _buffer(workBuffer.pointer, workBuffer.nRows, workBuffer.memorySpace, workBuffer.mathDomain);
MemoryTile _timeDiscretizer(timeDiscretizer.pointer, timeDiscretizer.nRows, timeDiscretizer.nCols, timeDiscretizer.memorySpace, timeDiscretizer.mathDomain);
// work out where to write the matrix-vector dot-product
MemoryBuffer *_out, *_in;
if (overwriteBuffer)
{
_out = &_buffer;
_in = &_solution;
}
else
{
_in = &_buffer;
_out = &_solution;
}
const ptr_t inPtr = _in->pointer;
const ptr_t outPtr = _out->pointer;
// multiplicate each solution with the respective time discretizer
for (unsigned i = 0; i < solution.nCols; ++i)
{
_buffer.pointer = workBuffer.pointer + i * _buffer.TotalSize();
_solution.pointer = solution.pointer + i * _solution.TotalSize();
_timeDiscretizer.pointer = timeDiscretizer.pointer + i * _timeDiscretizer.TotalSize();
_Dot(*_out, _timeDiscretizer, *_in);
}
// add the partial results into the latest solution
for (unsigned i = 1; i < solution.nCols; ++i)
{
// cumulative sum of each step contribution into the first column
_out->pointer = outPtr;
_in->pointer = outPtr + i * _in->TotalSize(); // re-use _in for convenience!
_AddEqual(*_out, *_in);
// copy the input solution into the older solution buffers
_out->pointer = _in->pointer;
_in->pointer = inPtr + i * _in->TotalSize();
_DeviceToDeviceCopy(*_out, *_in);
}
return cudaGetLastError();
}
int _SparseAdvance(MemoryBuffer& solution, SparseMemoryTile& timeDiscretizer, MemoryBuffer& workBuffer, const bool overwriteBuffer)
{
// work out where to write the matrix-vector dot-product
MemoryBuffer *_out, *_in;
if (overwriteBuffer)
{
_out = &workBuffer;
_in = &solution;
}
else
{
_in = &workBuffer;
_out = &solution;
}
_SparseDot(*_out, timeDiscretizer, *_in);
return cudaGetLastError();
}
int _MakeRungeKuttaGaussLegendre(const double dt,
const MemoryTile& spaceDiscretizer,
MemoryTile& timeDiscretizer)
{
constexpr double a00 = { .25 };
constexpr double sqrt3 = { 1.73205080756888 };
constexpr double a01 = { .25 - sqrt3 / 6.0 };
constexpr double a10 = { .25 + sqrt3 / 6.0 };
constexpr double a11 = { .25 };
MemoryTile eye(timeDiscretizer);
_Alloc(eye);
_Eye(eye);
MemoryTile A(timeDiscretizer);
_Alloc(A);
_Add(A, eye, spaceDiscretizer, -a00 * dt);
MemoryTile B(timeDiscretizer);
_Alloc(B);
_DeviceToDeviceCopy(B, spaceDiscretizer);
_Solve(A, B);
_Scale(B, a10 * dt);
MemoryTile C(timeDiscretizer);
_Alloc(C);
_DeviceToDeviceCopy(C, B);
_Scale(C, a01 * dt);
_AddEqualMatrix(C, eye, MatrixOperation::None, MatrixOperation::None, 1.0, a11 * dt);
MemoryTile C2(timeDiscretizer);
_Alloc(C2);
_DeviceToDeviceCopy(C2, C);
_Multiply(C, spaceDiscretizer, C2, MatrixOperation::None, MatrixOperation::None);
_Free(C2);
MemoryTile D(timeDiscretizer);
_Alloc(D);
_Add(D, C, eye, -1);
MemoryTile E(timeDiscretizer);
_Alloc(E);
_Add(E, eye, B);
MemoryTile k_2(timeDiscretizer);
_Alloc(k_2);
_Multiply(k_2, spaceDiscretizer, E, MatrixOperation::None, MatrixOperation::None);
_Solve(D, k_2);
MemoryTile F(timeDiscretizer);
_Alloc(F);
_Add(F, eye, k_2, a01 * dt);
MemoryTile k_1(timeDiscretizer);
_Alloc(k_1);
_Multiply(k_1, spaceDiscretizer, F, MatrixOperation::None, MatrixOperation::None);
_Solve(A, k_1);
_Eye(timeDiscretizer);
_AddEqualMatrix(k_1, k_2);
_AddEqualMatrix(timeDiscretizer, k_1, MatrixOperation::None, MatrixOperation::None, 1.0, .5 * dt);
_Free(eye);
_Free(A);
_Free(B);
_Free(C);
_Free(D);
_Free(E);
_Free(F);
_Free(k_1);
_Free(k_2);
return cudaGetLastError();
}
}
EXTERN_C
{
EXPORT int _MakeTimeDiscretizerAdvectionDiffusion(MemoryCube& timeDiscretizer, const MemoryTile& spaceDiscretizer, const SolverType solverType, const double dt)
{
MemoryTile _timeDiscretizer;
ExtractMatrixBufferFromCube(_timeDiscretizer, timeDiscretizer, 0);
switch (solverType)
{
case SolverType::ExplicitEuler:
// A = I + L * dt
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
_AddEqualMatrix(_timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, dt);
break;
case SolverType::ImplicitEuler:
// A = (I - L * dt)^(-1)
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
_AddEqualMatrix(_timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -dt);
_Invert(_timeDiscretizer);
break;
case SolverType::CrankNicolson:
{
// A = (I - L * .5 * dt)^(-1) * (I + L * .5 * dt)
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
// copy timeDiscretizer into leftOperator volatile buffer
MemoryTile leftOperator(_timeDiscretizer);
_Alloc(leftOperator);
_DeviceToDeviceCopy(leftOperator, _timeDiscretizer);
// left and right operator
_AddEqualMatrix(leftOperator, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -.5 * dt); // A = I - .5 * dt
_AddEqualMatrix(timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, .5 * dt); // B = I + .5 * dt
_Solve(leftOperator, _timeDiscretizer);
_Free(leftOperator);
}
break;
case SolverType::RungeKuttaRalston:
assert(timeDiscretizer.nCubes == 1);
detail::_MakeRungeKuttaDiscretizer<2>({ 0,
2.0 / 3.0, 0 },
{ .25, .75 }, dt, spaceDiscretizer, _timeDiscretizer);
break;
case SolverType::RungeKutta3:
assert(timeDiscretizer.nCubes == 1);
detail::_MakeRungeKuttaDiscretizer<3>({ 0,
.5, .0,
-1, 2, 0 },
{ 1.0 / 6.0, 2.0 / 3.0, 1.0 / 6.0 }, dt, spaceDiscretizer, _timeDiscretizer);
break;
case SolverType::RungeKutta4:
assert(timeDiscretizer.nCubes == 1);
detail::_MakeRungeKuttaDiscretizer<4>({ 0,
.5, .0,
0, .5, 0,
0, 0, 1, 0 },
{ 1.0 / 6.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 6.0 }, dt, spaceDiscretizer, _timeDiscretizer);
break;
case SolverType::RungeKuttaThreeEight:
assert(timeDiscretizer.nCubes == 1);
detail::_MakeRungeKuttaDiscretizer<4>({ 0,
1.0 / 3.0, .0,
-1.0 / 3.0, 1, 0,
1, -1, 1, 0 },
{ 1.0 / 8.0, 3.0 / 8.0, 3.0 / 8.0, 1.0 / 8.0 }, dt, spaceDiscretizer, _timeDiscretizer);
break;
case SolverType::RungeKuttaGaussLegendre4:
assert(timeDiscretizer.nCubes == 1);
detail::_MakeRungeKuttaGaussLegendre(dt, spaceDiscretizer, _timeDiscretizer);
break;
case SolverType::RichardsonExtrapolation2:
{
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
_AddEqualMatrix(_timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -dt);
_Invert(_timeDiscretizer);
_Scale(timeDiscretizer, -1.0);
MemoryTile halfIteration(_timeDiscretizer);
_Alloc(halfIteration);
_Eye(halfIteration);
_AddEqualMatrix(halfIteration, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -.5 * dt);
MemoryTile halfIterationSquared(_timeDiscretizer);
_Alloc(halfIterationSquared);
_Multiply(halfIterationSquared, halfIteration, halfIteration, MatrixOperation::None, MatrixOperation::None);
_Invert(halfIterationSquared);
_AddEqualMatrix(_timeDiscretizer, halfIterationSquared, MatrixOperation::None, MatrixOperation::None, 1.0, 2.0);
_Free(halfIteration);
_Free(halfIterationSquared);
}
break;
case SolverType::RichardsonExtrapolation3:
{
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
_AddEqualMatrix(_timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -dt);
_Invert(_timeDiscretizer);
// - F
_Scale(_timeDiscretizer, -1.0);
MemoryTile halfIteration(_timeDiscretizer);
_Alloc(halfIteration);
_Eye(halfIteration);
_AddEqualMatrix(halfIteration, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -.5 * dt);
MemoryTile halfIterationSquared(_timeDiscretizer);
_Alloc(halfIterationSquared);
_Multiply(halfIterationSquared, halfIteration, halfIteration, MatrixOperation::None, MatrixOperation::None);
_Invert(halfIterationSquared); // H
MemoryTile quarterIteration(_timeDiscretizer);
_Alloc(quarterIteration);
_Eye(quarterIteration);
_AddEqualMatrix(quarterIteration, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -.25 * dt);
MemoryTile quarterIterationFour(_timeDiscretizer);
_Alloc(quarterIterationFour);
_Multiply(halfIteration, quarterIteration, quarterIteration, MatrixOperation::None, MatrixOperation::None); // re-use halfIteration for convenience
_Multiply(quarterIterationFour, halfIteration, halfIteration, MatrixOperation::None, MatrixOperation::None);
_Invert(quarterIterationFour); // Q
// 2 * H - F
_AddEqualMatrix(_timeDiscretizer, halfIterationSquared, MatrixOperation::None, MatrixOperation::None, 1.0, 2.0);
// -(2 * H - F) / 3
_Scale(_timeDiscretizer, -1.0 / 3.0);
// 2 * Q - H
_Scale(halfIterationSquared, -1);
_AddEqualMatrix(halfIterationSquared, quarterIterationFour, MatrixOperation::None, MatrixOperation::None, 1.0, 2.0);
// (2 * Q - H) * 4/3 - (2 * H - F) / 3
_AddEqualMatrix(_timeDiscretizer, halfIterationSquared, MatrixOperation::None, MatrixOperation::None, 1.0, 4.0 / 3.0);
_Free(halfIteration);
_Free(halfIterationSquared);
_Free(quarterIteration);
_Free(quarterIterationFour);
}
break;
case SolverType::AdamsBashforth2:
// A_{n + 1} = (I + L * 1.5 * dt)
assert(timeDiscretizer.nCubes == 2);
_Eye(_timeDiscretizer);
_AddEqual(_timeDiscretizer, spaceDiscretizer, 1.5 * dt); // A = I + 1.5 * dt
// A_{n} = - L * .5 * dt
_timeDiscretizer.pointer += _timeDiscretizer.nRows * _timeDiscretizer.nCols * _timeDiscretizer.ElementarySize();
_DeviceToDeviceCopy(_timeDiscretizer, spaceDiscretizer);
_Scale(_timeDiscretizer, -.5 * dt);
break;
case SolverType::AdamsMouldon2:
{
// A_{n + 1} = (I - L * 5 / 12 * dt)^(-1) * (I + L * 2.0 / 3.0 * dt)
assert(timeDiscretizer.nCubes == 2);
// copy timeDiscretizer into leftOperator volatile buffer
MemoryTile leftOperator(_timeDiscretizer);
_Alloc(leftOperator);
_Eye(leftOperator);
_AddEqual(leftOperator, spaceDiscretizer, -5.0 / 12.0 * dt); // A = I - .5 * dt
_Eye(_timeDiscretizer);
_AddEqual(_timeDiscretizer, spaceDiscretizer, 2.0 / 3.0 * dt); // A = I - .5 * dt
_Solve(leftOperator, _timeDiscretizer);
// A_{n} = (I - L * 5 / 12 * dt)^(-1) * (- L * 1.0 / 12.0 * dt)
_timeDiscretizer.pointer += _timeDiscretizer.nRows * _timeDiscretizer.nCols * _timeDiscretizer.ElementarySize();
_DeviceToDeviceCopy(_timeDiscretizer, spaceDiscretizer);
_Scale(_timeDiscretizer, -1.0 / 12.0 * dt);
_Solve(leftOperator, _timeDiscretizer);
}
break;
default:
return CudaKernelException::_NotImplementedException;
}
return cudaGetLastError();
}
EXPORT int _MakeTimeDiscretizerWaveEquation(MemoryCube& timeDiscretizer, const MemoryTile& spaceDiscretizer, const SolverType solverType, const double dt)
{
MemoryTile _timeDiscretizer;
ExtractMatrixBufferFromCube(_timeDiscretizer, timeDiscretizer, 0);
switch (solverType)
{
case SolverType::ExplicitEuler:
{
// A = I
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
break;
}
case SolverType::ImplicitEuler:
{
// A = (I - L * dt^2)^(-1)
assert(timeDiscretizer.nCubes == 1);
_Eye(_timeDiscretizer);
_AddEqualMatrix(_timeDiscretizer, spaceDiscretizer, MatrixOperation::None, MatrixOperation::None, 1.0, -dt * dt);
_Invert(_timeDiscretizer);
break;
}
default:
return CudaKernelException::_NotImplementedException;
}
return cudaGetLastError();
}
}